(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 226984, 8433]*) (*NotebookOutlinePosition[ 227900, 8463]*) (* CellTagsIndexPosition[ 227856, 8459]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Math 325: Differential Equations\tAssignment 1 Solutions\t\tFall \ 1997\ \>", "Subsection"], Cell[CellGroupData[{ Cell["Problem 1", "Subsubsection"], Cell[TextData[{ "Verify that the functions\n", StyleBox[ "\ty1[t_] = 1; y2[t_] = t;\n\ty3[t_] = E^(-t); y4[t_] = t*E^(-t);\n", "Input"], "are solutions of the differential equation\n", StyleBox["\ty''''[t] + 2y'''[t] + y''[t] == 0\n", "Input"], "and calculate their Wronskian." }], "Text"], Cell["Solution", "Subsubsection"], Cell[TextData[{ "L[y_] := ", StyleBox["D[y,{t,4}] + 2 D[y,{t,3}] + D[y,{t,2}]", "Input"], ";" }], "Input"] }, Open ]], Cell["Clear[y1,y2,y3,y4]", "Input"], Cell[TextData[StyleBox[ "y1[t_] = 1; y2[t_] = t;\ny3[t_] = E^(-t); y4[t_] = t*E^(-t);", "Input"]], "Input"], Cell[CellGroupData[{ Cell["{L[y1[t]],L[y2[t]],L[y3[t]],L[y4[t]]}//Simplify", "Input"], Cell[BoxData[ \({0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ w[f1_,f2_,f3_,f4_] := Det[ {{ f1, f2, f3, f4 }, {D[f1,t], D[f2,t], D[f3,t], D[f4,t]}, {D[f1,{t,2}],D[f2,{t,2}],D[f3,{t,2}],D[f4,{t,2}]}, {D[f1,{t,3}],D[f2,{t,3}],D[f3,{t,3}],D[f4,{t,3}]}}];\ \>", "Input"], Cell[CellGroupData[{ Cell["w[y1[t],y2[t],y3[t],y4[t]]", "Input"], Cell[BoxData[ \(E\^\(\(-2\)\ t\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "Subsubsection"], Cell[TextData[{ "Show that the Wronskian of ", StyleBox["Cos[t]", "Input"], ", ", StyleBox["Cos[t]^3", "Input"], ", ", StyleBox["Cos[t]^5", "Input"], ", and ", StyleBox["Cos[5t]", "Input"], " is ", StyleBox["0", "Input"], ".\nEstablish this result without direct evaluation of the Wronskian." }], "Text"], Cell["Solution", "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell["w[Cos[t],Cos[t]^3,Cos[t]^5,Cos[5t]]", "Input"], Cell[BoxData[ \(\(-1200\)\ Cos[t]\^7\ Cos[5\ t]\ Sin[t]\^2 + 2400\ Cos[t]\^5\ Cos[5\ t]\ Sin[t]\^4 - 240\ Cos[t]\^3\ Cos[5\ t]\ Sin[t]\^6 + 240\ Cos[t]\^8\ Sin[t]\ Sin[5\ t] - 2400\ Cos[t]\^6\ Sin[t]\^3\ Sin[5\ t] + 1200\ Cos[t]\^4\ Sin[t]\^5\ Sin[5\ t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Simplify[%]", "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[TextData[{ "To prove that these functions are linearly dependent, and hence the \ Wronskian must be identically zero, use trig identities like\n", StyleBox["\tCos[A+B] -> Cos[A]Cos[B] - Sin[A]Sin[B];\n\t", "Input"], StyleBox["Sin[A+B] -> Sin[A]Cos[B] + Cos[A]Sin[B];", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "\tCos[5t] == Cos[2t]Cos[3t] - Sin[2t]Sin[3t]", "Input"]], "Input"], Cell[BoxData[ \(Cos[5\ t] == Cos[2\ t]\ Cos[3\ t] - Sin[2\ t]\ Sin[3\ t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["%", "Input"], " /.{Cos[3t]->Cos[t]Cos[2t] - Sin[t]Sin[2t],\n \t ", StyleBox["Sin[3t]", "Input"], "->Sin[t]Cos[2t] + Cos[t]Sin[2t]} //ExpandAll" }], "Input"], Cell[BoxData[ \(Cos[5\ t] == Cos[t]\ Cos[2\ t]\^2 - 2\ Cos[2\ t]\ Sin[t]\ Sin[2\ t] - Cos[t]\ Sin[2\ t]\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ % /.{Cos[2t]->Cos[t]^2-Sin[t]^2, Sin[2t]->2Sin[t]Cos[t]}//ExpandAll\ \ \>", "Input"], Cell[BoxData[ \(Cos[5\ t] == Cos[t]\^5 - 10\ Cos[t]\^3\ Sin[t]\^2 + 5\ Cos[t]\ Sin[t]\^4\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ % /.{Sin[t]^2->1-Cos[t]^2, Sin[t]^4->(1- \ Cos[t]^2)2}//ExpandAll\ \>", "Input"], Cell[BoxData[ \(Cos[5\ t] == 10\ Cos[t] - 20\ Cos[t]\^3 + 11\ Cos[t]\^5\)], "Output"] }, Open ]], Cell["Thus, we get the linear dependence relation:", "Text"], Cell["Cos[5t] - 10 Cos[t] + 20 Cos[t]^3 - 11 Cos[t]^5 == 0", "Input"], Cell[CellGroupData[{ Cell["Problem 3", "Subsubsection"], Cell[TextData[{ "Find all the 10th roots of ", StyleBox["1024", "Input"], ", including complex numbers. Convert these to points in the plane and plot \ them." }], "Text"], Cell["Solution", "Subsubsection"], Cell[CellGroupData[{ Cell["Solve[z^10 == 1024]", "Input"], Cell[BoxData[ \({{z \[Rule] \(-2\)}, {z \[Rule] 2}, { z \[Rule] \(-\(1\/2\)\) - \@5\/2 - I\ \@\(1\/2\ \((5 - \@5)\)\)}, { z \[Rule] 1\/2 + \@5\/2 - I\ \@\(1\/2\ \((5 - \@5)\)\)}, { z \[Rule] \(-\(1\/2\)\) - \@5\/2 + I\ \@\(1\/2\ \((5 - \@5)\)\)}, { z \[Rule] 1\/2 + \@5\/2 + I\ \@\(1\/2\ \((5 - \@5)\)\)}, { z \[Rule] 1\/2 - \@5\/2 - I\ \@\(1\/2\ \((5 + \@5)\)\)}, { z \[Rule] \(-\(1\/2\)\) + \@5\/2 - I\ \@\(1\/2\ \((5 + \@5)\)\)}, { z \[Rule] 1\/2 - \@5\/2 + I\ \@\(1\/2\ \((5 + \@5)\)\)}, { z \[Rule] \(-\(1\/2\)\) + \@5\/2 + I\ \@\(1\/2\ \((5 + \@5)\)\)}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["pts = {Re[z],Im[z]}/.%", "Input"], Cell[BoxData[ \({{\(-2\), 0}, {2, 0}, {\(-\(1\/2\)\) - \@5\/2, \(-\@\(1\/2\ \((5 - \@5)\)\)\)}, {1\/2 + \@5\/2, \(-\@\(1\/2\ \((5 - \@5)\)\)\)}, {\(-\(1\/2\)\) - \@5\/2, \@\(1\/2\ \((5 - \@5)\)\)}, {1\/2 + \@5\/2, \@\(1\/2\ \((5 - \@5)\)\)}, {1\/2 - \@5\/2, \(-\@\(1\/2\ \((5 + \@5)\)\)\)}, {\(-\(1\/2\)\) + \@5\/2, \(-\@\(1\/2\ \((5 + \@5)\)\)\)}, {1\/2 - \@5\/2, \@\(1\/2\ \((5 + \@5)\)\)}, {\(-\(1\/2\)\) + \@5\/2, \@\(1\/2\ \((5 + \@5)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["ListPlot[pts,AspectRatio->1]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.238095 0.5 0.250348 [ [.02381 .4875 -6 -9 ] [.02381 .4875 6 0 ] [.2619 .4875 -6 -9 ] [.2619 .4875 6 0 ] [.7381 .4875 -3 -9 ] [.7381 .4875 3 0 ] [.97619 .4875 -3 -9 ] [.97619 .4875 3 0 ] [.4875 .12448 -24 -4.5 ] [.4875 .12448 0 4.5 ] [.4875 .24965 -12 -4.5 ] [.4875 .24965 0 4.5 ] [.4875 .37483 -24 -4.5 ] [.4875 .37483 0 4.5 ] [.4875 .62517 -18 -4.5 ] [.4875 .62517 0 4.5 ] [.4875 .75035 -6 -4.5 ] [.4875 .75035 0 4.5 ] [.4875 .87552 -18 -4.5 ] [.4875 .87552 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .5 m .02381 .50625 L s [(-2)] .02381 .4875 0 1 Mshowa .2619 .5 m .2619 .50625 L s [(-1)] .2619 .4875 0 1 Mshowa .7381 .5 m .7381 .50625 L s [(1)] .7381 .4875 0 1 Mshowa .97619 .5 m .97619 .50625 L s [(2)] .97619 .4875 0 1 Mshowa .125 Mabswid .07143 .5 m .07143 .50375 L s .11905 .5 m .11905 .50375 L s .16667 .5 m .16667 .50375 L s .21429 .5 m .21429 .50375 L s .30952 .5 m .30952 .50375 L s .35714 .5 m .35714 .50375 L s .40476 .5 m .40476 .50375 L s .45238 .5 m .45238 .50375 L s .54762 .5 m .54762 .50375 L s .59524 .5 m .59524 .50375 L s .64286 .5 m .64286 .50375 L s .69048 .5 m .69048 .50375 L s .78571 .5 m .78571 .50375 L s .83333 .5 m .83333 .50375 L s .88095 .5 m .88095 .50375 L s .92857 .5 m .92857 .50375 L s .25 Mabswid 0 .5 m 1 .5 L s .5 .12448 m .50625 .12448 L s [(-1.5)] .4875 .12448 1 0 Mshowa .5 .24965 m .50625 .24965 L s [(-1)] .4875 .24965 1 0 Mshowa .5 .37483 m .50625 .37483 L s [(-0.5)] .4875 .37483 1 0 Mshowa .5 .62517 m .50625 .62517 L s [(0.5)] .4875 .62517 1 0 Mshowa .5 .75035 m .50625 .75035 L s [(1)] .4875 .75035 1 0 Mshowa .5 .87552 m .50625 .87552 L s [(1.5)] .4875 .87552 1 0 Mshowa .125 Mabswid .5 .02434 m .50375 .02434 L s .5 .04937 m .50375 .04937 L s .5 .07441 m .50375 .07441 L s .5 .09944 m .50375 .09944 L s .5 .14951 m .50375 .14951 L s .5 .17455 m .50375 .17455 L s .5 .19958 m .50375 .19958 L s .5 .22462 m .50375 .22462 L s .5 .27469 m .50375 .27469 L s .5 .29972 m .50375 .29972 L s .5 .32476 m .50375 .32476 L s .5 .34979 m .50375 .34979 L s .5 .39986 m .50375 .39986 L s .5 .4249 m .50375 .4249 L s .5 .44993 m .50375 .44993 L s .5 .47497 m .50375 .47497 L s .5 .52503 m .50375 .52503 L s .5 .55007 m .50375 .55007 L s .5 .5751 m .50375 .5751 L s .5 .60014 m .50375 .60014 L s .5 .65021 m .50375 .65021 L s .5 .67524 m .50375 .67524 L s .5 .70028 m .50375 .70028 L s .5 .72531 m .50375 .72531 L s .5 .77538 m .50375 .77538 L s .5 .80042 m .50375 .80042 L s .5 .82545 m .50375 .82545 L s .5 .85049 m .50375 .85049 L s .5 .90056 m .50375 .90056 L s .5 .92559 m .50375 .92559 L s .5 .95063 m .50375 .95063 L s .5 .97566 m .50375 .97566 L s .25 Mabswid .5 0 m .5 1 L s 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath .008 w .02381 .5 Mdot .97619 .5 Mdot .11475 .2057 Mdot .88525 .2057 Mdot .11475 .7943 Mdot .88525 .7943 Mdot .35285 .02381 Mdot .64715 .02381 Mdot .35285 .97619 Mdot .64715 .97619 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLc0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0 002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0 002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000SP0008l0 00?o0000SP0008l000?o0000SP0007X00ol400;o0`03o`H000?o0000SP0007/000?o00000`02o`H0 00?o00000`000ol0002>0000M005o`8000?o00002`000ol0000300?oSP0007/000?o00002003o`H0 00?o0000SP0007/000?o000020000ol000060003o`0008h0001j00;o2P04o`D000?o0000SP0008l0 00?o0000SP0008l000?o0000SP0008l000?o0000SP0008l00_n?0000S`000ol0002>0000S`000ol0 002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0 002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0 002?0003o`0008h0002?0003o`0008h0002?00;oS`0002000_m]0003o`0006/00_lQ00008002ofd0 00?o0000J`02ob40002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0 002?00;oS`0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008H0 0ol60003o`0008h000270003o`0000D000?o0000SP0008001Ol20003o`0000D00on>0000Q`000ol0 00050003o`0008h000270003o`0000D000?o0000SP0008H00_l70003o`0008h0002?0003o`0008h0 002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000SP0008l0 00?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l0 0_n?0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0 002>0000S`000ol0002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0 002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l0 00?o0000SP0008l000?o0000SP0008l000?o0000SP0007/00_l400;o0`03o`H000?o0000SP0007X0 00Co003o0`02o`H000?o00000`000ol0002>0000M005o`050?l00?l02`000ol0000300?oSP0007X0 00Co003o2003o`H000?o0000SP0007X000Co003o20000ol000060003o`0008h0001k00;o2@04o`D0 00?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l00_n?0000S`000ol0 002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0 002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0 002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l0 00?o0000SP0000L01?m100?o@0000ol0000o00?o@P04o`H000080003o`00048000?o0000?`000ol0 00100003o`00048000?o00001P0000030?oo00?o0`000ol0000j00Go0P000ol0000o0003o`000400 00?o0000@`000ol0000500002P000ol000100003o`0003l000?o0000@0000ol000140003o`0000@0 00070004o`00od8000?o0000?`02od4000?o0000@@001?l00?l600002002od800_m10003o`0003l0 0_m400;o1`0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l0 00?o0000SP000?oo8Ol000H00_l<0003o`0000/000?o00002P000ol0000;0003o`0000/000?o0000 2P000ol0000;0003o`0000/000?o00002P000ol0000;0003o`0000/000?o00002P000ol0000;0003 o`0000/000?o00002P000ol0000;0003o`0000/000?o00002P000ol0000;0003o`0000X00_l70000 1P000ol000120003o`00044000?o0000@@000ol000120003o`0000D0002?0003o`0008h0002?0003 o`0008h0002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l00_n?0000 S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000 S`000ol0002>0000S`000ol0002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003 o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000 SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0007/00_l400;o0`03o`H000?o0000 SP0007X000Co003o0`02o`H000?o00000`000ol0002>0000NP001?l00?l;0003o`0000<00on>0000 NP001?l00?l800?o1P000ol0002>0000NP001?l00?l80003o`0000H000?o0000SP0007/00_l900Co 1@000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`02ohl0002?0003 o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003 o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0008l000?o0000SP0008l00_n?0000S`000ol0002>0000S`000ol0002>0000 S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000 S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002600?o 1P000ol0002>0000Q`000ol000050003o`0008h000270003o`0000D00on>0000Q`000ol000050003 o`0008h000270003o`0000D000?o0000SP0008H00_l70003o`0008h0002?0003o`0008h0002?0003 o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0002000_m]0003o`0006/00_lQ00008002ofd000?o0000J`02ob40002?00;o S`0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0008l00_n?0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000 S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`02ohl0002?0003 o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0001j00?o1002o`<00ol60003 o`0008h0001k0003o`0000<00_l60003o`0000<000?o0000SP0007/000?o00002`000ol0000300?o SP0007/000?o00002003o`H000?o0000SP0007/000?o000020000ol000060003o`0008h0001j00;o 2P04o`D000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l00_n?0000 S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000S`000ol0002>0000 S`000ol0002>0000S`02ohl0002?0003o`0008h0002?0003o`0008h0002?0003o`0008h0002?0003 o`0008h0002?0003o`0008h0002?0003o`0008h0002?00;oS`0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP0006@00_lY0003 o`0002L00_mU0000I002obT00_lX00;oI@0008l000?o0000SP0008l000?o0000SP0008l000?o0000 SP0008l000?o0000SP0008l000?o0000SP0008l000?o0000SP000001\ \>"], ImageRangeCache->{{{0, 287}, {287, 0}} -> {-2.10002, -1.99724, 0.0146343, 0.013918}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "Another method: use polar coordinates ", StyleBox["{r Cos[t], r Sin[t]}", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["N[1024^(1/10)]", "Input"], Cell[BoxData[ \(2.`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["pts = Table[{2 Cos[2Pi k/10], 2 Sin[2Pi k/10]}, {k,0,9}]", "Input"], Cell[BoxData[ \({{2, 0}, {1\/2\ \((1 + \@5)\), \@\(1\/2\ \((5 - \@5)\)\)}, { 1\/2\ \((\(-1\) + \@5)\), \@\(1\/2\ \((5 + \@5)\)\)}, { 1\/2\ \((1 - \@5)\), \@\(1\/2\ \((5 + \@5)\)\)}, { 1\/2\ \((\(-1\) - \@5)\), \@\(1\/2\ \((5 - \@5)\)\)}, {\(-2\), 0}, { 1\/2\ \((\(-1\) - \@5)\), \(-\@\(1\/2\ \((5 - \@5)\)\)\)}, { 1\/2\ \((1 - \@5)\), \(-\@\(1\/2\ \((5 + \@5)\)\)\)}, { 1\/2\ \((\(-1\) + \@5)\), \(-\@\(1\/2\ \((5 + \@5)\)\)\)}, { 1\/2\ \((1 + \@5)\), \(-\@\(1\/2\ \((5 - \@5)\)\)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "Find the roots of the characteristic equation and the corrseponding \ fundamental solutions to the differential equation\n", StyleBox[ "\ty''''[t] + 6y'''[t] + 17y''[t] + 22y'[t] + 14y[t] == 0\n", "Input"], "Superimpose plots of the fundamental solutions.Then find and plot the \ particular solution that satisfies the initial conditions:\n", StyleBox["\ty[0] == 1, y'[0] == 1, y''[0]==1, y'''[0] == 1\n", "Input"], "Compare your solution with the solution given by ", StyleBox["DSolve[]", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell[CellGroupData[{ Cell["Solve[r^4 + 6r^3 + 17r^2 + 22r + 14 == 0]", "Input"], Cell[BoxData[ \({{r \[Rule] \(-1\) - I}, {r \[Rule] \(-1\) + I}, { r \[Rule] \(-2\) - I\ \@3}, {r \[Rule] \(-2\) + I\ \@3}}\)], "Output"] }, Open ]] }, Open ]], Cell["The corresponding fundamental solutions are:", "Text"], Cell["\<\ y1 = E^(-t)Cos[t]; y2 = E^(-t)Sin[t]; y3 = E^(-2t)Cos[Sqrt[3]t]; y4 = E^(-2t)Sin[Sqrt[3]t];\ \>", "Input"], Cell[CellGroupData[{ Cell["Plot[{y1,y2,y3,y4}, {t,0,2Pi}, PlotRange->{-0.2,1.1}]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.0950822 0.475411 [ [.17539 .08258 -3 -9 ] [.17539 .08258 3 0 ] [.32696 .08258 -3 -9 ] [.32696 .08258 3 0 ] [.47854 .08258 -3 -9 ] [.47854 .08258 3 0 ] [.63011 .08258 -3 -9 ] [.63011 .08258 3 0 ] [.78169 .08258 -3 -9 ] [.78169 .08258 3 0 ] [.93327 .08258 -3 -9 ] [.93327 .08258 3 0 ] [.01131 0 -24 -4.5 ] [.01131 0 0 4.5 ] [.01131 .19016 -18 -4.5 ] [.01131 .19016 0 4.5 ] [.01131 .28525 -18 -4.5 ] [.01131 .28525 0 4.5 ] [.01131 .38033 -18 -4.5 ] [.01131 .38033 0 4.5 ] [.01131 .47541 -18 -4.5 ] [.01131 .47541 0 4.5 ] [.01131 .57049 -6 -4.5 ] [.01131 .57049 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .09508 m .17539 .10133 L s [(1)] .17539 .08258 0 1 Mshowa .32696 .09508 m .32696 .10133 L s [(2)] .32696 .08258 0 1 Mshowa .47854 .09508 m .47854 .10133 L s [(3)] .47854 .08258 0 1 Mshowa .63011 .09508 m .63011 .10133 L s [(4)] .63011 .08258 0 1 Mshowa .78169 .09508 m .78169 .10133 L s [(5)] .78169 .08258 0 1 Mshowa .93327 .09508 m .93327 .10133 L s [(6)] .93327 .08258 0 1 Mshowa .125 Mabswid .05412 .09508 m .05412 .09883 L s .08444 .09508 m .08444 .09883 L s .11476 .09508 m .11476 .09883 L s .14507 .09508 m .14507 .09883 L s .2057 .09508 m .2057 .09883 L s .23602 .09508 m .23602 .09883 L s .26633 .09508 m .26633 .09883 L s .29665 .09508 m .29665 .09883 L s .35728 .09508 m .35728 .09883 L s .38759 .09508 m .38759 .09883 L s .41791 .09508 m .41791 .09883 L s .44822 .09508 m .44822 .09883 L s .50885 .09508 m .50885 .09883 L s .53917 .09508 m .53917 .09883 L s .56948 .09508 m .56948 .09883 L s .5998 .09508 m .5998 .09883 L s .66043 .09508 m .66043 .09883 L s .69074 .09508 m .69074 .09883 L s .72106 .09508 m .72106 .09883 L s .75138 .09508 m .75138 .09883 L s .81201 .09508 m .81201 .09883 L s .84232 .09508 m .84232 .09883 L s .87264 .09508 m .87264 .09883 L s .90295 .09508 m .90295 .09883 L s .96358 .09508 m .96358 .09883 L s .9939 .09508 m .9939 .09883 L s .25 Mabswid 0 .09508 m 1 .09508 L s .02381 0 m .03006 0 L s [(-0.2)] .01131 0 1 0 Mshowa .02381 .19016 m .03006 .19016 L s [(0.2)] .01131 .19016 1 0 Mshowa .02381 .28525 m .03006 .28525 L s [(0.4)] .01131 .28525 1 0 Mshowa .02381 .38033 m .03006 .38033 L s [(0.6)] .01131 .38033 1 0 Mshowa .02381 .47541 m .03006 .47541 L s [(0.8)] .01131 .47541 1 0 Mshowa .02381 .57049 m .03006 .57049 L s [(1)] .01131 .57049 1 0 Mshowa .125 Mabswid .02381 .02377 m .02756 .02377 L s .02381 .04754 m .02756 .04754 L s .02381 .07131 m .02756 .07131 L s .02381 .11885 m .02756 .11885 L s .02381 .14262 m .02756 .14262 L s .02381 .16639 m .02756 .16639 L s .02381 .21393 m .02756 .21393 L s .02381 .23771 m .02756 .23771 L s .02381 .26148 m .02756 .26148 L s .02381 .30902 m .02756 .30902 L s .02381 .33279 m .02756 .33279 L s .02381 .35656 m .02756 .35656 L s .02381 .4041 m .02756 .4041 L s .02381 .42787 m .02756 .42787 L s .02381 .45164 m .02756 .45164 L s .02381 .49918 m .02756 .49918 L s .02381 .52295 m .02756 .52295 L s .02381 .54672 m .02756 .54672 L s .02381 .59426 m .02756 .59426 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .57049 m .06244 .45162 L .10458 .33542 L .14415 .24575 L .18221 .179 L .20342 .14979 L .22272 .1278 L .24299 .10902 L .26171 .09521 L .28086 .08421 L .29159 .07927 L .30164 .07538 L .32037 .0698 L .3303 .06764 L .34099 .06585 L .3511 .06463 L .35681 .06412 L .36212 .06375 L .36475 .06361 L .36722 .0635 L .36947 .06341 L .37185 .06334 L .37308 .06331 L .37444 .06328 L .37569 .06326 L .37683 .06324 L .37747 .06324 L .37817 .06323 L .37891 .06323 L .37961 .06322 L .38027 .06322 L .38096 .06322 L .38221 .06322 L .38335 .06323 L .3846 .06324 L .38589 .06325 L .3871 .06327 L .38927 .06331 L .39158 .06337 L .39688 .06355 L .40169 .06376 L .41109 .06432 L .41997 .06499 L .45843 .06907 L .49935 .07445 L .53875 .07969 L .57664 .08424 L .61698 .08828 L .6558 .09129 L .67555 .09249 L Mistroke .69708 .09359 L .71795 .09444 L .73684 .09505 L .75655 .09554 L .76749 .09576 L .77754 .09593 L .79682 .09618 L .80696 .09627 L .81766 .09635 L .82716 .0964 L .83249 .09642 L .83734 .09643 L .84185 .09644 L .84424 .09645 L .84676 .09645 L .84897 .09645 L .85014 .09646 L .8514 .09646 L .85255 .09646 L .85361 .09646 L .85462 .09646 L .85568 .09646 L .85692 .09646 L .85762 .09646 L .85826 .09646 L .85945 .09646 L .8607 .09646 L .86182 .09646 L .86287 .09646 L .86525 .09646 L .86783 .09645 L .87018 .09645 L .87546 .09644 L .8849 .09642 L .89384 .09639 L .91373 .09631 L .93214 .09622 L .97137 .096 L .97619 .09597 L Mfstroke .02381 .09508 m .04262 .14705 L .06244 .18798 L .07286 .20446 L .08255 .21703 L .0932 .22801 L .10458 .23683 L .1148 .24242 L .12056 .2447 L .12589 .24628 L .13054 .24728 L .13318 .2477 L .13566 .248 L .137 .24812 L .13827 .24821 L .13941 .24827 L .14065 .24832 L .14136 .24834 L .142 .24835 L .14274 .24835 L .14344 .24835 L .14468 .24833 L .14538 .24831 L .14603 .24829 L .14729 .24822 L .14848 .24815 L .15072 .24795 L .15314 .24768 L .15576 .2473 L .16048 .24644 L .16485 .24543 L .17487 .24243 L .18551 .23833 L .22326 .21849 L .26347 .19289 L .30215 .16821 L .34329 .14472 L .38292 .12612 L .42103 .1123 L .44228 .10626 L .46159 .10172 L .4819 .09784 L .50063 .095 L .51982 .09273 L .53056 .09172 L .54061 .09092 L .55938 .08978 L .56933 .08934 L .58002 .08898 L .59015 .08873 L Mistroke .59588 .08863 L .60119 .08856 L .60383 .08853 L .6063 .08851 L .60855 .08849 L .61093 .08848 L .61217 .08847 L .61353 .08847 L .61476 .08846 L .61591 .08846 L .61661 .08846 L .61727 .08846 L .61851 .08846 L .6192 .08846 L .61985 .08846 L .62055 .08846 L .62129 .08846 L .62244 .08846 L .62369 .08846 L .62499 .08847 L .6262 .08847 L .63069 .0885 L .63326 .08851 L .63558 .08853 L .6408 .08858 L .65022 .0887 L .6591 .08885 L .69762 .0897 L .73859 .09083 L .77804 .09191 L .81598 .09286 L .85637 .09369 L .89525 .09431 L .91504 .09456 L .93658 .09478 L .97619 .09508 L Mfstroke .02381 .57049 m .04262 .45748 L .06244 .35325 L .08426 .26013 L .10458 .19391 L .12297 .14955 L .14264 .11603 L .15285 .1034 L .16394 .0928 L .17441 .08533 L .18408 .0803 L .18874 .07843 L .19375 .07677 L .19808 .07561 L .20281 .07461 L .20567 .07412 L .20833 .07375 L .21108 .07343 L .21358 .07321 L .21603 .07304 L .21835 .07293 L .21962 .07289 L .22079 .07286 L .22208 .07284 L .22272 .07283 L .22343 .07283 L .22474 .07283 L .22594 .07284 L .22657 .07285 L .22727 .07287 L .22867 .07291 L .22991 .07295 L .23123 .07301 L .23364 .07313 L .23849 .07347 L .24294 .07388 L .25313 .07508 L .26386 .07663 L .30197 .08298 L .32322 .08629 L .34253 .08886 L .36284 .09105 L .38157 .09261 L .40076 .09379 L .41086 .09426 L .42155 .09467 L .43099 .09495 L .44113 .09519 L .4507 .09536 L .45944 .09548 L Mistroke .46422 .09553 L .46931 .09557 L .47371 .0956 L .47846 .09563 L .48139 .09564 L .48407 .09565 L .48659 .09566 L .48935 .09567 L .49055 .09567 L .49183 .09567 L .49294 .09567 L .49414 .09567 L .49485 .09567 L .4955 .09567 L .49679 .09567 L .49796 .09567 L .4992 .09567 L .50034 .09567 L .50159 .09567 L .5029 .09567 L .50411 .09567 L .50628 .09567 L .5086 .09567 L .5139 .09566 L .51871 .09564 L .537 .09558 L .57704 .0954 L .61556 .09525 L .63547 .0952 L .65653 .09515 L .66599 .09513 L .67614 .09512 L .69447 .0951 L .70462 .09509 L .71568 .09508 L .72613 .09507 L .73078 .09507 L .7358 .09507 L .74081 .09507 L .74541 .09507 L .75008 .09507 L .75444 .09507 L .75922 .09507 L .76176 .09507 L .76446 .09507 L .76583 .09507 L .76731 .09507 L .76871 .09507 L .77 .09507 L .77116 .09507 L Mistroke .77244 .09507 L .77309 .09507 L .77378 .09507 L .77503 .09507 L .77572 .09507 L .77636 .09507 L .77757 .09507 L .77887 .09507 L .78028 .09507 L .78281 .09507 L .78514 .09507 L .79051 .09507 L .79618 .09507 L .81627 .09507 L .85427 .09507 L .87543 .09508 L .89471 .09508 L .91494 .09508 L .93364 .09508 L .95274 .09508 L .96281 .09508 L .9735 .09508 L .97619 .09508 L Mfstroke .02381 .09508 m .03279 .13832 L .04262 .17419 L .04758 .18829 L .05288 .20073 L .05785 .21015 L .06244 .21709 L .06788 .22335 L .07075 .22586 L .07379 .22798 L .07642 .22939 L .07889 .23038 L .0803 .23081 L .08159 .23112 L .08231 .23125 L .08309 .23137 L .08382 .23146 L .08451 .23153 L .08577 .23159 L .08648 .23159 L .08713 .23158 L .08833 .2315 L .0896 .23137 L .09074 .23119 L .0918 .23098 L .09422 .23037 L .09684 .22948 L .09922 .22848 L .10458 .22567 L .11449 .21872 L .12366 .21081 L .14443 .1901 L .18369 .15086 L .22144 .12216 L .24079 .11177 L .25075 .10749 L .26165 .10356 L .28169 .09815 L .29262 .09604 L .30279 .09453 L .31331 .09335 L .31848 .0929 L .32324 .09255 L .33216 .09205 L .33686 .09186 L .34183 .0917 L .34436 .09164 L .34676 .09158 L .34894 .09155 L .35127 .09151 L Mistroke .35259 .0915 L .35378 .09149 L .355 .09148 L .35612 .09147 L .35743 .09146 L .35812 .09146 L .35884 .09146 L .36006 .09145 L .36076 .09145 L .3614 .09145 L .36254 .09145 L .36378 .09146 L .36495 .09146 L .36603 .09147 L .36808 .09148 L .37025 .0915 L .37475 .09154 L .37972 .09161 L .38984 .0918 L .3991 .09202 L .41997 .09257 L .45965 .09362 L .47966 .09406 L .49781 .09439 L .51746 .09466 L .52837 .09478 L .53842 .09487 L .55765 .09501 L .56776 .09506 L .57846 .0951 L .58792 .09513 L .59313 .09514 L .59807 .09515 L .60266 .09516 L .60765 .09516 L .61186 .09517 L .6164 .09517 L .61892 .09517 L .62124 .09517 L .62359 .09518 L .62578 .09518 L .62819 .09518 L .62947 .09518 L .63083 .09518 L .63152 .09518 L .63226 .09518 L .63292 .09518 L .63361 .09518 L .63493 .09518 L .63615 .09518 L Mistroke .63745 .09518 L .63863 .09518 L .63929 .09518 L .64001 .09518 L .64131 .09518 L .64382 .09518 L .64614 .09518 L .65147 .09518 L .65715 .09517 L .67857 .09516 L .69875 .09515 L .73732 .09512 L .75688 .09511 L .77834 .0951 L .79878 .09509 L .81785 .09509 L .82831 .09508 L .83822 .09508 L .84704 .09508 L .85678 .09508 L .86644 .09508 L .87138 .09508 L .8767 .09508 L .88166 .09508 L .88621 .09508 L .89055 .09508 L .89513 .09508 L .8977 .09508 L .90014 .09508 L .90232 .09508 L .90468 .09508 L .90591 .09508 L .90726 .09508 L .90794 .09508 L .90867 .09508 L .90999 .09508 L .9113 .09508 L .91255 .09508 L .91367 .09508 L .91488 .09508 L .91613 .09508 L .91731 .09508 L .9195 .09508 L .92442 .09508 L .92867 .09508 L .93326 .09508 L .95325 .09508 L .97433 .09508 L .97619 .09508 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLc00;o10000ol0000Q00;o [@0001/000Ko003o0?lE0003o`0000d000?o00000`000ol0000P00;o[`0001/000Ko003o0?lD0003 o`0000d000?o00000`000ol0000O00;o/@0001/000Co003o0P000ol0000A0003o`0000d000?o0000 0`000ol0000M00;o/`0001/000Oo0000o`3o018000?o00003@000ol000030003o`0001d000?o0000 /`0001/000Go0000o`020003o`0000l000?o000030000ol000030003o`0001`00_nf00006`02o`80 00Co003o4@000ol0000;0003o`0000<000?o00006`02okP0000K0003o`00008000Co003o3`000ol0 000;0003o`0000@000?o00006@02okX0000K0003o`00008000Co003o3`000ol0000:0003o`0000@0 00?o00006002ok`0000K0003o`00008000Co003o3P000ol0000:0003o`0000@000?o000060000ol0 002l00006`000ol000020005o`000?l03@000ol000090003o`0000@000?o00005`02okl0000800;o 1002o`<01?l40003o`0000<000Co003o30000ol000090003o`0000D000?o00005@02ol4000070004 o`00o`<00_l40003o`0000@000?o00000`001Ol0003o00/000?o000020000ol000050003o`0001D0 00?o0000`@0000L000Co003o2P000ol0000300;o10001Ol0003o00/000?o00001P02o`P000?o0000 4`02ol@000070004o`00o`/000?o00000P000ol000040005o`000?l02@000ol000060003o`0000L0 00?o00004`000ol0003400001`001?l00?l80004o`00o`@000?o00001@001Ol0003o00P000?o0000 1@000ol000070003o`0001800_o700002002o`X00_l50003o`0000D000?o00000P000ol000050003 o`0000@000?o000020000ol0000@00;ob@0001/000?o00001P001Ol0003o00H000?o000010000ol0 00080003o`00010000?o0000b@0001/000?o00001P000ol000020003o`0000<000?o00000`000ol0 00090003o`0000h00_o<00006`02o`L000?o00000P000ol000030003o`00008000?o00002@000ol0 000>0003o`000<`0000K0003o`0000L00_l30005o`000?l00`02o`/000?o00003@02oll0000K0003 o`0000T000Go0000o`020004o`00o`d000?o00002`02om40000K0003o`0000T000?o00000P001Ol0 o`3o00d000?o00002P02om<0000K0003o`0000X00ol200?o3P000ol0000800;oe@0001/000?o0000 3004o`030?l000`000?o00001`02omL0000K00;o40000ol00002o`X000?o00001P02omT0000K0003 o`0000l000?o00000P000ol000070003o`0000D000?o0000f@0001/000?o00003P000ol0000400?o 1@000ol0000200Cog00001/000?o00003P000ol0000700[oh00001/000?o00003P000ol0000;0003 o`000><0000K0003o`0000d000?o000030000ol0003S00006`000ol0000=0003o`0000/000?o0000 i00001/00_l>0003o`0000/000?o0000i00001/000?o00003@000ol0000:0003o`000>D0000K0003 o`0000`000?o00002`000ol0003U00006`000ol0000<0003o`0000/000?o0000i@0000P00_l400;o 1003o`@000?o000030000ol0000:0003o`000>H000070004o`00o`<00_l50003o`0000<000?o0000 30000ol0000:0003o`000>H000070004o`00o`P01?l400;o30000ol0000:0003o`000>L000070004 o`00o`P000?o0?l01@000ol0000;0003o`0000X000?o0000i`0000L000Co003o20000ol0o`050003 o`0000/000?o00002@000ol0003X00002002o`X00_l50003o`0000/000?o00002@000ol0003X0000 6`000ol0000:0003o`0000T000?o0000j@0001/000?o00002P000ol000090003o`000>T0000K00;o 2`000ol000080003o`000>X0000K0003o`0000X000?o000020000ol0003Z00006`000ol0000:0003 o`0000P000?o0000jP0001/000?o00002@000ol000080003o`000>/0000K0003o`0000T000?o0000 20000ol0003[00006`000ol000090003o`0000L000?o0000k00001/000?o00002@000ol000070003 o`000>`0000K00;o2@000ol000070003o`000>d0000K0003o`0000P000?o00001`000ol0003]0000 6`000ol000080003o`0000H000?o0000kP0001/000?o000020000ol000060003o`000>h0000K0003 o`0000L000?o00001`000ol0003^00006`000ol000070003o`0000H000?o0000k`0001/00_l80003 o`0000H000?o0000k`0001/000?o00001`000ol000060003o`000>l0000K0003o`0000L000?o0000 1@000ol0003`00006`000ol000060003o`0000H000?o0000l00000P00_l400;o1002o`D000?o0000 1P000ol000060003o`000?0000070004o`00o`<00_l30004o`00o`@000?o00001P000ol000050003 o`000?4000070004o`00o`P000Co003o1002o`L000?o00001@000ol0003a00001`001?l00?l800?o 1@000ol000060003o`0000D000?o0000l@0000L000Co003o20000ol000050003o`0000D000?o0000 1@000ol0003b00002002o`X00ol40003o`0000D000?o00001@000ol0003b00006`000ol000050003 o`0000@000?o0000l`0001/000?o00001@000ol000040003o`000?<0000K0003o`0000D000?o0000 10000ol0003c00006`02o`H000?o00000`000ol0003d00006`000ol000040003o`0000@000?o0000 m00001/000?o000010000ol000040003o`000?@0000K0003o`0000@000?o00000`000ol0003e0000 6`000ol000040003o`0000<000?o0000m@0001/000?o000010000ol000030003o`000?D0000K00;o 1@000ol000020003o`000?H0000K0003o`0000<000?o00000`000ol0003f00006`000ol000030003 o`0000<000?o0000mP0001/000?o00000`000ol000020003o`000?L0000K0003o`0000<000?o0000 0P000ol0003g00006`000ol000030003o`00008000?o0000m`0001/00_l30003o`00008000?o0000 n00001/000?o00000P000ol000020003o`000?P0000K0003o`00008000?o00000P000ol0003h0000 6`000ol000020005o`000?l0n`0001/000?o00000P001Ol0003o0?/0000800;o1002o`@00_l50003 o`00008000Go0000o`3k00001`001?l00?l300;o0`001?l00?l40005o`000?l00`000ol0003j0000 1`001?l00?l80004o`00o`@00_l20005o`000?l0o00000L000Co003o2@02o`D000Go0000o`030003 o`000?X000070004o`00o`P000Co003o10001Ol0003o008000?o0000n`0000P00_l:00;o1@001Ol0 003o008000?o0000n`0001/000Go0000o`020003o`000?/0000K0004o`00o`8000?o0000o00001/0 0_l01@3o003o0?h0000K0004o`00o`8000?o0000o00001/000Ko003o0?oo00006`001_l00?l0ool0 000K0006o`00o`3oo`0001/000?o0?l00P000ol0003m00006`03o`030?l00?l0000K0005o`3o0?l0 o`0100006`001Ol0o`3o0?l00@0001/000Co0?ooo`0200006`001?l0oooo0080000K00;o00<0o`00 o`0100006`03ool00`0001/00ooo00<0000K00?oo`0300006`02ool0100001/00_oo00@0000C00?o 1@02ool0100001@000?o000010000ol0003o00<0000D0003o`0000@00_oo00@0000D0003o`0000@0 00?o0000o`03000050000ol000040003o`000?l00`0001<00_l60003o`000?l00`0001/000?o0000 o`0300006`000ol0003o00<0000K00;oo`0400006`000ol0003o00<0000K0003o`000?l00`0001/0 00?o0000o`0300006`000ol0003o00<0000K0003o`000?l00`0001/000?o0000o`030000o`0Q0000 o`0Q0000o`0Q0000o`0Q0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -0.267364, 0.0248015, 0.00790751}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "To find the particular solution ", StyleBox["phi[t]", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell["phi[t_] = c1 y1 + c2 y2 + c3 y3 + c4 y4", "Input"], Cell[BoxData[ \(c1\ E\^\(-t\)\ Cos[t] + c3\ E\^\(\(-2\)\ t\)\ Cos[\@3\ t] + c2\ E\^\(-t\)\ Sin[t] + c4\ E\^\(\(-2\)\ t\)\ Sin[\@3\ t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["eqns = {phi[0]==1,phi'[0]==1,phi''[0]==1,phi'''[0]==1}", "Input"], Cell[BoxData[ \({c1 + c3 == 1, \(-c1\) + c2 - 2\ c3 + \@3\ c4 == 1, \(-2\)\ c2 + c3 - 4\ \@3\ c4 == 1, 2\ c1 + 2\ c2 + 10\ c3 + 9\ \@3\ c4 == 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[eqns]", "Input"], Cell[BoxData[ \({{c2 \[Rule] 96\/13, c4 \[Rule] \(-\(\(15\ \@3\)\/13\)\), c1 \[Rule] \(-\(12\/13\)\), c3 \[Rule] 25\/13}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["phi[t] /. First[%]", "Input"], Cell[BoxData[ \(\(-\(12\/13\)\)\ E\^\(-t\)\ Cos[t] + 25\/13\ E\^\(\(-2\)\ t\)\ Cos[\@3\ t] + 96\/13\ E\^\(-t\)\ Sin[t] - 15\/13\ \@3\ E\^\(\(-2\)\ t\)\ Sin[\@3\ t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plot[%, {t,0,2Pi}]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.0432612 0.311849 [ [.17539 .03076 -3 -9 ] [.17539 .03076 3 0 ] [.32696 .03076 -3 -9 ] [.32696 .03076 3 0 ] [.47854 .03076 -3 -9 ] [.47854 .03076 3 0 ] [.63011 .03076 -3 -9 ] [.63011 .03076 3 0 ] [.78169 .03076 -3 -9 ] [.78169 .03076 3 0 ] [.93327 .03076 -3 -9 ] [.93327 .03076 3 0 ] [.01131 .12122 -24 -4.5 ] [.01131 .12122 0 4.5 ] [.01131 .19919 -18 -4.5 ] [.01131 .19919 0 4.5 ] [.01131 .27715 -24 -4.5 ] [.01131 .27715 0 4.5 ] [.01131 .35511 -6 -4.5 ] [.01131 .35511 0 4.5 ] [.01131 .43307 -24 -4.5 ] [.01131 .43307 0 4.5 ] [.01131 .51104 -18 -4.5 ] [.01131 .51104 0 4.5 ] [.01131 .589 -24 -4.5 ] [.01131 .589 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .04326 m .17539 .04951 L s [(1)] .17539 .03076 0 1 Mshowa .32696 .04326 m .32696 .04951 L s [(2)] .32696 .03076 0 1 Mshowa .47854 .04326 m .47854 .04951 L s [(3)] .47854 .03076 0 1 Mshowa .63011 .04326 m .63011 .04951 L s [(4)] .63011 .03076 0 1 Mshowa .78169 .04326 m .78169 .04951 L s [(5)] .78169 .03076 0 1 Mshowa .93327 .04326 m .93327 .04951 L s [(6)] .93327 .03076 0 1 Mshowa .125 Mabswid .05412 .04326 m .05412 .04701 L s .08444 .04326 m .08444 .04701 L s .11476 .04326 m .11476 .04701 L s .14507 .04326 m .14507 .04701 L s .2057 .04326 m .2057 .04701 L s .23602 .04326 m .23602 .04701 L s .26633 .04326 m .26633 .04701 L s .29665 .04326 m .29665 .04701 L s .35728 .04326 m .35728 .04701 L s .38759 .04326 m .38759 .04701 L s .41791 .04326 m .41791 .04701 L s .44822 .04326 m .44822 .04701 L s .50885 .04326 m .50885 .04701 L s .53917 .04326 m .53917 .04701 L s .56948 .04326 m .56948 .04701 L s .5998 .04326 m .5998 .04701 L s .66043 .04326 m .66043 .04701 L s .69074 .04326 m .69074 .04701 L s .72106 .04326 m .72106 .04701 L s .75138 .04326 m .75138 .04701 L s .81201 .04326 m .81201 .04701 L s .84232 .04326 m .84232 .04701 L s .87264 .04326 m .87264 .04701 L s .90295 .04326 m .90295 .04701 L s .96358 .04326 m .96358 .04701 L s .9939 .04326 m .9939 .04701 L s .25 Mabswid 0 .04326 m 1 .04326 L s .02381 .12122 m .03006 .12122 L s [(0.25)] .01131 .12122 1 0 Mshowa .02381 .19919 m .03006 .19919 L s [(0.5)] .01131 .19919 1 0 Mshowa .02381 .27715 m .03006 .27715 L s [(0.75)] .01131 .27715 1 0 Mshowa .02381 .35511 m .03006 .35511 L s [(1)] .01131 .35511 1 0 Mshowa .02381 .43307 m .03006 .43307 L s [(1.25)] .01131 .43307 1 0 Mshowa .02381 .51104 m .03006 .51104 L s [(1.5)] .01131 .51104 1 0 Mshowa .02381 .589 m .03006 .589 L s [(1.75)] .01131 .589 1 0 Mshowa .125 Mabswid .02381 .05885 m .02756 .05885 L s .02381 .07445 m .02756 .07445 L s .02381 .09004 m .02756 .09004 L s .02381 .10563 m .02756 .10563 L s .02381 .13682 m .02756 .13682 L s .02381 .15241 m .02756 .15241 L s .02381 .168 m .02756 .168 L s .02381 .18359 m .02756 .18359 L s .02381 .21478 m .02756 .21478 L s .02381 .23037 m .02756 .23037 L s .02381 .24596 m .02756 .24596 L s .02381 .26156 m .02756 .26156 L s .02381 .29274 m .02756 .29274 L s .02381 .30833 m .02756 .30833 L s .02381 .32393 m .02756 .32393 L s .02381 .33952 m .02756 .33952 L s .02381 .3707 m .02756 .3707 L s .02381 .3863 m .02756 .3863 L s .02381 .40189 m .02756 .40189 L s .02381 .41748 m .02756 .41748 L s .02381 .44867 m .02756 .44867 L s .02381 .46426 m .02756 .46426 L s .02381 .47985 m .02756 .47985 L s .02381 .49544 m .02756 .49544 L s .02381 .52663 m .02756 .52663 L s .02381 .54222 m .02756 .54222 L s .02381 .55781 m .02756 .55781 L s .02381 .57341 m .02756 .57341 L s .02381 .02767 m .02756 .02767 L s .02381 .01208 m .02756 .01208 L s .02381 .60459 m .02756 .60459 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .35511 m .06244 .4431 L .08255 .49083 L .10458 .53784 L .11448 .55577 L .12507 .57216 L .13499 .58457 L .14415 .59332 L .14911 .59693 L .15437 .59988 L .15709 .60105 L .15964 .60193 L .16245 .60265 L .16402 .60293 L .16477 .60305 L .16546 .60313 L .1667 .60324 L .16802 .60331 L .16924 .60332 L .17039 .60328 L .17105 .60325 L .17174 .60319 L .17298 .60305 L .17432 .60285 L .17578 .60256 L .17823 .60192 L .18086 .60103 L .1856 .59889 L .19088 .59569 L .19577 .59201 L .20684 .5812 L .22696 .55368 L .26622 .47796 L .30398 .39058 L .34418 .29635 L .38287 .21449 L .42401 .14295 L .44453 .11398 L .46363 .09103 L .50174 .05601 L .5222 .04253 L .54078 .03307 L .55997 .02575 L .57079 .02259 L .58077 .02023 L .58981 .01851 L .59953 .01706 L .6051 .01641 L .61019 .01591 L .61502 .01553 L Mistroke .62017 .01521 L .62284 .01507 L .62535 .01497 L .62779 .01489 L .63004 .01482 L .63218 .01478 L .63327 .01476 L .63444 .01474 L .63561 .01473 L .63671 .01472 L .63801 .01472 L .63919 .01472 L .64048 .01472 L .64113 .01472 L .64185 .01472 L .64311 .01473 L .6443 .01475 L .64697 .01479 L .64985 .01485 L .65511 .01502 L .65993 .01521 L .66976 .01575 L .67889 .01638 L .69961 .01821 L .7387 .02258 L .78025 .02762 L .82029 .03216 L .85881 .0359 L .89978 .03904 L .92017 .04029 L .93923 .04127 L .97619 .04273 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLc00006`000ol0001`0003o`0008l0 000K00;oL0000ol0002@00006`000ol0001_0003o`000900000200;o1002o`<01?l200?o1@000ol0 001^0003o`00094000001@3o003o00<00_l40003o`0000D000?o00000P000ol0001]0003o`000980 00001@3o003o00X000?o000010000ol0000200;oKP000ol0002B000000D0o`00o`0;0003o`0000?o 1@000ol0001/0003o`0009<000001@3o003o00P000Co003o0P000ol000050003o`0006/000?o0000 U00000800_l:00;o0`04o`@000?o0000JP000ol0002E00006`000ol0001Y0003o`0009H0000K00;o J@000ol0002G00006`000ol0001W0003o`0009P0000K0003o`0006L000?o0000V00001/000?o0000 IP000ol0002I00006`02ofL000?o0000V@0001/000?o0000I@000ol0002J00006`000ol0001T0003 o`0009/0000K0003o`0006@000?o0000V`0001/00_mT0003o`0009`0000K0003o`0006<000?o0000 W00001/000?o0000HP000ol0002M00006`000ol0001Q0003o`0009h0000K00;oHP000ol0002N0000 6`000ol0001P0003o`0009l0000800;o1002o`<00ol50003o`00060000?o0000W`0000L000Co003o 0`02o`H000?o00000P000ol0001O0003o`000:0000070004o`00o`/000?o00000P02of0000?o0000 X00000L000Co003o2003o`D000?o0000GP000ol0002Q00001`001?l00?l80003o`0000D000?o0000 G@000ol0002R00002002o`T01?l40003o`0005d000?o0000XP0001/00_mM0003o`000:<0000K0003 o`0005`000?o0000X`0001/000?o0000F`000ol0002T00006`000ol0001K0003o`000:@0000K0003 o`0005X000?o0000Y@0001/00_mK0003o`000:D0000K0003o`0005T000?o0000YP0001/000?o0000 F@000ol0002V00006`000ol0001I0003o`000:H0000K00;oF@000ol0002W00006`000ol0001H0003 o`000:L0000K0003o`0005L000?o0000Z00001/000?o0000E`000ol0002X00006`02oeL000?o0000 Z@0001/000?o0000EP000ol0002Y00000P02o`@00_l40003o`0000800ol50003o`0005D000?o0000 ZP0000050?l00?l00`02o`@000?o00001@000ol000020003o`0005D000?o0000ZP0000050?l00?l0 2P000ol000040003o`0000800_mE0003o`000:/000001@3o003o00X000Co00000ol50003o`0005@0 00?o0000Z`0000050?l00?l02`001?l00?l70003o`0005@000?o0000Z`0000800_l900Co0P04o`@0 00?o0000D`000ol0002/00006`02oe@000?o0000[00001/000?o0000DP000ol0002]00006`000ol0 001B0003o`000:d0000K0003o`00054000?o0000[P0001/00_mB0003o`000:h0000K0003o`000500 00?o0000[`0001/000?o0000D0000ol0002_00006`000ol0001@0003o`000:l0000K00;oD0000ol0 002`00006`000ol0001?0003o`000;00000K0003o`0004h000?o0000/@0001/000?o0000CP000ol0 002a00006`000ol0001=0003o`000;80000K00;oCP000ol0002b00006`000ol0001<0003o`000;<0 000C00?o1@000ol0001<0003o`000;<0000D0003o`0000@000?o0000C0000ol0002c000050000ol0 000400;oC0000ol0002d000050000ol000040003o`0004/000?o0000]00001@000?o00001002od/0 00?o0000]@0001<00_l600;oB`000ol0002e00006`03odT000?o0000]P0001/000?o0?l0B@000ol0 002f00006`000ol0o`190003o`000;H0000K0004o`00odL000?o0000]`0001/00_l00`3o00160003 o`000;L0000K0005o`000?l0A@000ol0002h00006`001Ol0003o04D000?o0000^00001/000?o0000 0P000ol000110003o`000;T0000K00;o0`000ol000110003o`000;T0000K0003o`00008000?o0000 @0000ol0002j00006`000ol000030003o`0003l000?o0000^P0001/000?o00000`000ol0000o0003 o`000;X0000K00;o1@000ol0000m0003o`000;/0000K0003o`0000@000?o0000?@000ol0002k0000 00@0oooo1002o`<01?l200?o1@000ol000050003o`0003/000?o0000_000008000?o00000`02o`@0 00?o00001@000ol000020003o`0000D000?o0000>`000ol0002l00000P000ol0000:0003o`0000@0 00?o00000P02o`H000?o0000>P000ol0002m00000P000ol0000;0003o`0000?o1@000ol000060003 o`0003T000?o0000_@00008000?o000020001?l00?l20003o`0000D000?o00001P000ol0000i0003 o`000;d000000`3oo`0;00;o0`04o`@000?o00001`000ol0000g0003o`000;h0000K0003o`0000L0 00?o0000=`000ol0002n00006`02o`T000?o0000=@000ol0002o00006`000ol000080003o`0003D0 00?o0000_`0001/000?o000020000ol0000d0003o`000<00000K0003o`0000T000?o0000<`000ol0 003000006`02o`X000?o0000<`000ol0003000006`000ol0000:0003o`00034000?o0000`@0001/0 00?o00002P000ol0000a0003o`000<40000K0003o`0000X000?o0000<0000ol0003200006`02o``0 00?o0000;`000ol0003200006`000ol0000;0003o`0002h000?o0000``0001/000?o00002`000ol0 000^0003o`000<<0000K0003o`0000`000?o0000;0000ol0003400006`02o`d000?o0000;0000ol0 003400006`000ol0000=0003o`0002X000?o0000a@0000L00ol400;o0`03o`D000?o00003@000ol0 000Z0003o`00000006`000ol0000H0003o`0001D000?o0000c`0001/00_lI0003o`0001D000?o0000c`0001/0 00?o00006@000ol0000C0003o`000=000000103oool400;o10000ol0000200?o1@000ol0000I0003 o`00018000?o0000d@00008000?o00000`02o`@000?o00001@000ol000020003o`0001X00_lA0003 o`000=8000020003o`0000X000?o000010000ol0000200;o7@000ol0000=0003o`000=<000020003 o`0000X000Co00000ol50003o`0001d000?o00002P02omH000020003o`0000/000Co003o1`000ol0 000N00;o2@000ol0003F000000<0ool02P04o`801?l40003o`0002000_l400?of@0001/00_lS00Co g00001/000?o0000o`0300006`000ol0003o00<0000K0003o`000?l00`0001/000?o0000o`030000 o`0Q0000o`0Q0000o`0Q0000o`0Q0000o`0Q0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -0.219225, 0.0248015, 0.0120549}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "DSolve[{", StyleBox["y''''[t] + 6y'''[t] + 17y''[t] + 22y'[t] + 14y[t] == 0", "Input"], ",\n\t", StyleBox["y[0] == 1, y'[0] == 1, y''[0]==1, y'''[0] == 1", "Input"], "},y[t],t]" }], "Input"], Cell[BoxData[ \({{y[t] \[Rule] E\^\(-t\)\ \(( \(5\ \((\(-I\) + \@3)\)\ E\^\(t + \((\(-2\) - I\ \@3)\)\ t\)\)\/\(2\ I + 4\ \@3\) + 5\/26\ \((5 + 3\ I\ \@3)\)\ E\^\(t + \((\(-2\) + I\ \@3)\)\ t\) - \(12\ Cos[t]\)\/13 + \(96\ Sin[t]\)\/13)\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Plot[y[t]/.First[%],{t,0,2Pi}]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.0432612 0.311849 [ [.17539 .03076 -3 -9 ] [.17539 .03076 3 0 ] [.32696 .03076 -3 -9 ] [.32696 .03076 3 0 ] [.47854 .03076 -3 -9 ] [.47854 .03076 3 0 ] [.63011 .03076 -3 -9 ] [.63011 .03076 3 0 ] [.78169 .03076 -3 -9 ] [.78169 .03076 3 0 ] [.93327 .03076 -3 -9 ] [.93327 .03076 3 0 ] [.01131 .12122 -24 -4.5 ] [.01131 .12122 0 4.5 ] [.01131 .19919 -18 -4.5 ] [.01131 .19919 0 4.5 ] [.01131 .27715 -24 -4.5 ] [.01131 .27715 0 4.5 ] [.01131 .35511 -6 -4.5 ] [.01131 .35511 0 4.5 ] [.01131 .43307 -24 -4.5 ] [.01131 .43307 0 4.5 ] [.01131 .51104 -18 -4.5 ] [.01131 .51104 0 4.5 ] [.01131 .589 -24 -4.5 ] [.01131 .589 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .04326 m .17539 .04951 L s [(1)] .17539 .03076 0 1 Mshowa .32696 .04326 m .32696 .04951 L s [(2)] .32696 .03076 0 1 Mshowa .47854 .04326 m .47854 .04951 L s [(3)] .47854 .03076 0 1 Mshowa .63011 .04326 m .63011 .04951 L s [(4)] .63011 .03076 0 1 Mshowa .78169 .04326 m .78169 .04951 L s [(5)] .78169 .03076 0 1 Mshowa .93327 .04326 m .93327 .04951 L s [(6)] .93327 .03076 0 1 Mshowa .125 Mabswid .05412 .04326 m .05412 .04701 L s .08444 .04326 m .08444 .04701 L s .11476 .04326 m .11476 .04701 L s .14507 .04326 m .14507 .04701 L s .2057 .04326 m .2057 .04701 L s .23602 .04326 m .23602 .04701 L s .26633 .04326 m .26633 .04701 L s .29665 .04326 m .29665 .04701 L s .35728 .04326 m .35728 .04701 L s .38759 .04326 m .38759 .04701 L s .41791 .04326 m .41791 .04701 L s .44822 .04326 m .44822 .04701 L s .50885 .04326 m .50885 .04701 L s .53917 .04326 m .53917 .04701 L s .56948 .04326 m .56948 .04701 L s .5998 .04326 m .5998 .04701 L s .66043 .04326 m .66043 .04701 L s .69074 .04326 m .69074 .04701 L s .72106 .04326 m .72106 .04701 L s .75138 .04326 m .75138 .04701 L s .81201 .04326 m .81201 .04701 L s .84232 .04326 m .84232 .04701 L s .87264 .04326 m .87264 .04701 L s .90295 .04326 m .90295 .04701 L s .96358 .04326 m .96358 .04701 L s .9939 .04326 m .9939 .04701 L s .25 Mabswid 0 .04326 m 1 .04326 L s .02381 .12122 m .03006 .12122 L s [(0.25)] .01131 .12122 1 0 Mshowa .02381 .19919 m .03006 .19919 L s [(0.5)] .01131 .19919 1 0 Mshowa .02381 .27715 m .03006 .27715 L s [(0.75)] .01131 .27715 1 0 Mshowa .02381 .35511 m .03006 .35511 L s [(1)] .01131 .35511 1 0 Mshowa .02381 .43307 m .03006 .43307 L s [(1.25)] .01131 .43307 1 0 Mshowa .02381 .51104 m .03006 .51104 L s [(1.5)] .01131 .51104 1 0 Mshowa .02381 .589 m .03006 .589 L s [(1.75)] .01131 .589 1 0 Mshowa .125 Mabswid .02381 .05885 m .02756 .05885 L s .02381 .07445 m .02756 .07445 L s .02381 .09004 m .02756 .09004 L s .02381 .10563 m .02756 .10563 L s .02381 .13682 m .02756 .13682 L s .02381 .15241 m .02756 .15241 L s .02381 .168 m .02756 .168 L s .02381 .18359 m .02756 .18359 L s .02381 .21478 m .02756 .21478 L s .02381 .23037 m .02756 .23037 L s .02381 .24596 m .02756 .24596 L s .02381 .26156 m .02756 .26156 L s .02381 .29274 m .02756 .29274 L s .02381 .30833 m .02756 .30833 L s .02381 .32393 m .02756 .32393 L s .02381 .33952 m .02756 .33952 L s .02381 .3707 m .02756 .3707 L s .02381 .3863 m .02756 .3863 L s .02381 .40189 m .02756 .40189 L s .02381 .41748 m .02756 .41748 L s .02381 .44867 m .02756 .44867 L s .02381 .46426 m .02756 .46426 L s .02381 .47985 m .02756 .47985 L s .02381 .49544 m .02756 .49544 L s .02381 .52663 m .02756 .52663 L s .02381 .54222 m .02756 .54222 L s .02381 .55781 m .02756 .55781 L s .02381 .57341 m .02756 .57341 L s .02381 .02767 m .02756 .02767 L s .02381 .01208 m .02756 .01208 L s .02381 .60459 m .02756 .60459 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .35511 m .06244 .4431 L .08255 .49083 L .10458 .53784 L .11448 .55577 L .12507 .57216 L .13499 .58457 L .14415 .59332 L .14911 .59693 L .15437 .59988 L .15709 .60105 L .15964 .60193 L .16245 .60265 L .16402 .60293 L .16477 .60305 L .16546 .60313 L .1667 .60324 L .16802 .60331 L .16924 .60332 L .17039 .60328 L .17105 .60325 L .17174 .60319 L .17298 .60305 L .17432 .60285 L .17578 .60256 L .17823 .60192 L .18086 .60103 L .1856 .59889 L .19088 .59569 L .19577 .59201 L .20684 .5812 L .22696 .55368 L .26622 .47796 L .30398 .39058 L .34418 .29635 L .38287 .21449 L .42401 .14295 L .44453 .11398 L .46363 .09103 L .50174 .05601 L .5222 .04253 L .54078 .03307 L .55997 .02575 L .57079 .02259 L .58077 .02023 L .58981 .01851 L .59953 .01706 L .6051 .01641 L .61019 .01591 L .61502 .01553 L Mistroke .62017 .01521 L .62284 .01507 L .62535 .01497 L .62779 .01489 L .63004 .01482 L .63218 .01478 L .63327 .01476 L .63444 .01474 L .63561 .01473 L .63671 .01472 L .63801 .01472 L .63919 .01472 L .64048 .01472 L .64113 .01472 L .64185 .01472 L .64311 .01473 L .6443 .01475 L .64697 .01479 L .64985 .01485 L .65511 .01502 L .65993 .01521 L .66976 .01575 L .67889 .01638 L .69961 .01821 L .7387 .02258 L .78025 .02762 L .82029 .03216 L .85881 .0359 L .89978 .03904 L .92017 .04029 L .93923 .04127 L .97619 .04273 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLc00006`000ol0001`0003o`0008l0 000K00;oL0000ol0002@00006`000ol0001_0003o`000900000200;o1002o`<01?l200?o1@000ol0 001^0003o`00094000001@3o003o00<00_l40003o`0000D000?o00000P000ol0001]0003o`000980 00001@3o003o00X000?o000010000ol0000200;oKP000ol0002B000000D0o`00o`0;0003o`0000?o 1@000ol0001/0003o`0009<000001@3o003o00P000Co003o0P000ol000050003o`0006/000?o0000 U00000800_l:00;o0`04o`@000?o0000JP000ol0002E00006`000ol0001Y0003o`0009H0000K00;o J@000ol0002G00006`000ol0001W0003o`0009P0000K0003o`0006L000?o0000V00001/000?o0000 IP000ol0002I00006`02ofL000?o0000V@0001/000?o0000I@000ol0002J00006`000ol0001T0003 o`0009/0000K0003o`0006@000?o0000V`0001/00_mT0003o`0009`0000K0003o`0006<000?o0000 W00001/000?o0000HP000ol0002M00006`000ol0001Q0003o`0009h0000K00;oHP000ol0002N0000 6`000ol0001P0003o`0009l0000800;o1002o`<00ol50003o`00060000?o0000W`0000L000Co003o 0`02o`H000?o00000P000ol0001O0003o`000:0000070004o`00o`/000?o00000P02of0000?o0000 X00000L000Co003o2003o`D000?o0000GP000ol0002Q00001`001?l00?l80003o`0000D000?o0000 G@000ol0002R00002002o`T01?l40003o`0005d000?o0000XP0001/00_mM0003o`000:<0000K0003 o`0005`000?o0000X`0001/000?o0000F`000ol0002T00006`000ol0001K0003o`000:@0000K0003 o`0005X000?o0000Y@0001/00_mK0003o`000:D0000K0003o`0005T000?o0000YP0001/000?o0000 F@000ol0002V00006`000ol0001I0003o`000:H0000K00;oF@000ol0002W00006`000ol0001H0003 o`000:L0000K0003o`0005L000?o0000Z00001/000?o0000E`000ol0002X00006`02oeL000?o0000 Z@0001/000?o0000EP000ol0002Y00000P02o`@00_l40003o`0000800ol50003o`0005D000?o0000 ZP0000050?l00?l00`02o`@000?o00001@000ol000020003o`0005D000?o0000ZP0000050?l00?l0 2P000ol000040003o`0000800_mE0003o`000:/000001@3o003o00X000Co00000ol50003o`0005@0 00?o0000Z`0000050?l00?l02`001?l00?l70003o`0005@000?o0000Z`0000800_l900Co0P04o`@0 00?o0000D`000ol0002/00006`02oe@000?o0000[00001/000?o0000DP000ol0002]00006`000ol0 001B0003o`000:d0000K0003o`00054000?o0000[P0001/00_mB0003o`000:h0000K0003o`000500 00?o0000[`0001/000?o0000D0000ol0002_00006`000ol0001@0003o`000:l0000K00;oD0000ol0 002`00006`000ol0001?0003o`000;00000K0003o`0004h000?o0000/@0001/000?o0000CP000ol0 002a00006`000ol0001=0003o`000;80000K00;oCP000ol0002b00006`000ol0001<0003o`000;<0 000C00?o1@000ol0001<0003o`000;<0000D0003o`0000@000?o0000C0000ol0002c000050000ol0 000400;oC0000ol0002d000050000ol000040003o`0004/000?o0000]00001@000?o00001002od/0 00?o0000]@0001<00_l600;oB`000ol0002e00006`03odT000?o0000]P0001/000?o0?l0B@000ol0 002f00006`000ol0o`190003o`000;H0000K0004o`00odL000?o0000]`0001/00_l00`3o00160003 o`000;L0000K0005o`000?l0A@000ol0002h00006`001Ol0003o04D000?o0000^00001/000?o0000 0P000ol000110003o`000;T0000K00;o0`000ol000110003o`000;T0000K0003o`00008000?o0000 @0000ol0002j00006`000ol000030003o`0003l000?o0000^P0001/000?o00000`000ol0000o0003 o`000;X0000K00;o1@000ol0000m0003o`000;/0000K0003o`0000@000?o0000?@000ol0002k0000 00@0oooo1002o`<01?l200?o1@000ol000050003o`0003/000?o0000_000008000?o00000`02o`@0 00?o00001@000ol000020003o`0000D000?o0000>`000ol0002l00000P000ol0000:0003o`0000@0 00?o00000P02o`H000?o0000>P000ol0002m00000P000ol0000;0003o`0000?o1@000ol000060003 o`0003T000?o0000_@00008000?o000020001?l00?l20003o`0000D000?o00001P000ol0000i0003 o`000;d000000`3oo`0;00;o0`04o`@000?o00001`000ol0000g0003o`000;h0000K0003o`0000L0 00?o0000=`000ol0002n00006`02o`T000?o0000=@000ol0002o00006`000ol000080003o`0003D0 00?o0000_`0001/000?o000020000ol0000d0003o`000<00000K0003o`0000T000?o0000<`000ol0 003000006`02o`X000?o0000<`000ol0003000006`000ol0000:0003o`00034000?o0000`@0001/0 00?o00002P000ol0000a0003o`000<40000K0003o`0000X000?o0000<0000ol0003200006`02o``0 00?o0000;`000ol0003200006`000ol0000;0003o`0002h000?o0000``0001/000?o00002`000ol0 000^0003o`000<<0000K0003o`0000`000?o0000;0000ol0003400006`02o`d000?o0000;0000ol0 003400006`000ol0000=0003o`0002X000?o0000a@0000L00ol400;o0`03o`D000?o00003@000ol0 000Z0003o`00000006`000ol0000H0003o`0001D000?o0000c`0001/00_lI0003o`0001D000?o0000c`0001/0 00?o00006@000ol0000C0003o`000=000000103oool400;o10000ol0000200?o1@000ol0000I0003 o`00018000?o0000d@00008000?o00000`02o`@000?o00001@000ol000020003o`0001X00_lA0003 o`000=8000020003o`0000X000?o000010000ol0000200;o7@000ol0000=0003o`000=<000020003 o`0000X000Co00000ol50003o`0001d000?o00002P02omH000020003o`0000/000Co003o1`000ol0 000N00;o2@000ol0003F000000<0ool02P04o`801?l40003o`0002000_l400?of@0001/00_lS00Co g00001/000?o0000o`0300006`000ol0003o00<0000K0003o`000?l00`0001/000?o0000o`030000 o`0Q0000o`0Q0000o`0Q0000o`0Q0000o`0Q0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -0.219225, 0.0248015, 0.0120549}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "Find the general solution to the following differential equation using the \ method of undetermined coefficents.\n", StyleBox[ "\ty''''[t] + 2y'''[t] + 5y''[t] + 8y'[t] + 4y[t]==Sin[2t] + t*E^(-t)\n", "Input"], "Superimpose plots of the particular solutions satisfying\n", StyleBox["\ty[0] == y0, y'[0] == 0, y''[0] == 0, y'''[0] == 0\n", "Input"], "for several values of ", StyleBox["y0", "Input"], ". Explain why all these solutions seem to pass through the ", StyleBox["t", "Input"], "-axis at the same points and at regular intervals for large values of ", StyleBox["t", "Input"], ". Try to determine the length of this common interval." }], "Text"], Cell["Solution", "Subsubsection"], Cell[CellGroupData[{ Cell["Solve[r^4 + 2r^3 + 5r^2 + 8r + 4 == 0]", "Input"], Cell[BoxData[ \({{r \[Rule] \(-1\)}, {r \[Rule] \(-1\)}, {r \[Rule] \(-2\)\ I}, { r \[Rule] 2\ I}}\)], "Output"] }, Open ]] }, Open ]], Cell["The general solution to the homogeneous equation is:", "Text"], Cell["y[t_] = c1 E^(-t) + c2 t*E^(-t) + c3 Cos[2t] + c4 Sin[2t];", "Input"], Cell["\<\ A solution to the non-homogeneous equation should have the \ form\ \>", "Text"], Cell[TextData[{ StyleBox["Y[t_] = t(a Cos[2t] + b Sin[2t]) + t^2(c*t + d)E^", "Input"], "(-t)", StyleBox[";", "Input"] }], "Input"], Cell[TextData[{ "The usual guesses must be multiplied by powers of ", StyleBox["t", "Input"], " for otherwise they would satisfy the homogeneous equation (and give ", StyleBox["0", "Input"], " instead of the desired function). To find the correct values of the \ constants, we plug into the equation." }], "Text"], Cell["L[y_] := D[y,{t,4}]+2D[y,{t,3}]+5D[y,{t,2}]+8D[y,t]+4y;", "Input"], Cell[CellGroupData[{ Cell[TextData[{ "L[Y[t]] == (", StyleBox["Sin[2t] + t*E^", "Input"], "(-t)", StyleBox[")//ExpandAll", "Input"] }], "Input"], Cell[BoxData[ \(\(-12\)\ c\ E\^\(-t\) + 10\ d\ E\^\(-t\) + 30\ c\ E\^\(-t\)\ t - 16\ a\ Cos[2\ t] - 12\ b\ Cos[2\ t] + 12\ a\ Sin[2\ t] - 16\ b\ Sin[2\ t] == E\^\(-t\)\ t + Sin[2\ t]\)], "Output"] }, Open ]], Cell["Equating coefficients of like terms we get", "Text"], Cell["\<\ eqns = {-12c + 10d == 0, 30c == 1, -16a - 12b == 0, 12a - 16b == 1};\ \>", "Input"], Cell[CellGroupData[{ Cell["sol = Solve[eqns]", "Input"], Cell[BoxData[ \({{a \[Rule] 3\/100, b \[Rule] \(-\(1\/25\)\), c \[Rule] 1\/30, d \[Rule] 1\/25}}\)], "Output"] }, Open ]], Cell["\<\ The general solution to the non-homogeneous equation is then\ \>", "Text"], Cell[CellGroupData[{ Cell["phi[t_] = y[t] + Y[t] /. First[sol]", "Input"], Cell[BoxData[ \(c1\ E\^\(-t\) + c2\ E\^\(-t\)\ t + E\^\(-t\)\ \((1\/25 + t\/30)\)\ t\^2 + c3\ Cos[2\ t] + t\ \((3\/100\ Cos[2\ t] - 1\/25\ Sin[2\ t])\) + c4\ Sin[2\ t]\)], "Output"] }, Open ]], Cell["Let's check it is a solution:", "Text"], Cell[CellGroupData[{ Cell["L[phi[t]]//Simplify", "Input"], Cell[BoxData[ \(E\^\(-t\)\ t + Sin[2\ t]\)], "Output"] }, Open ]], Cell["Particular solutions:", "Text"], Cell[CellGroupData[{ Cell["eqns = {phi[0]==y0, phi'[0]==0, phi''[0]==0, phi'''[0]==0}", "Input"], Cell[BoxData[ \({c1 + c3 == y0, 3\/100 - c1 + c2 + 2\ c4 == 0, \(-\(2\/25\)\) + c1 - 2\ c2 - 4\ c3 == 0, \(-\(2\/5\)\) - c1 + 3\ c2 - 8\ c4 == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["sol = Solve[eqns,{c1,c2,c3,c4}]", "Input"], Cell[BoxData[ \({{c1 \[Rule] 28\/625\ \((1 + 25\ y0)\), c2 \[Rule] 1\/125\ \((9 + 100\ y0)\), c3 \[Rule] 1\/625\ \((\(-28\) - 75\ y0)\), c4 \[Rule] \(\(-143\) + 800\ y0\)\/5000}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y[y0_,t_] = phi[t] /. First[sol]", "Input"], Cell[BoxData[ \(E\^\(-t\)\ \((1\/25 + t\/30)\)\ t\^2 + 28\/625\ E\^\(-t\)\ \((1 + 25\ y0)\) + 1\/125\ E\^\(-t\)\ t\ \((9 + 100\ y0)\) + 1\/625\ \((\(-28\) - 75\ y0)\)\ Cos[2\ t] + t\ \((3\/100\ Cos[2\ t] - 1\/25\ Sin[2\ t])\) + \(\((\(-143\) + 800\ y0)\)\ Sin[2\ t]\)\/5000\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Plot[Evaluate[Table[y[y0,t],{y0,0,5}]],{t,0,15}, PlotRange->{-1,5}]\ \>", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0634921 0.103006 0.103006 [ [.15079 .09051 -3 -9 ] [.15079 .09051 3 0 ] [.27778 .09051 -3 -9 ] [.27778 .09051 3 0 ] [.40476 .09051 -3 -9 ] [.40476 .09051 3 0 ] [.53175 .09051 -3 -9 ] [.53175 .09051 3 0 ] [.65873 .09051 -6 -9 ] [.65873 .09051 6 0 ] [.78571 .09051 -6 -9 ] [.78571 .09051 6 0 ] [.9127 .09051 -6 -9 ] [.9127 .09051 6 0 ] [.01131 0 -12 -4.5 ] [.01131 0 0 4.5 ] [.01131 .20601 -6 -4.5 ] [.01131 .20601 0 4.5 ] [.01131 .30902 -6 -4.5 ] [.01131 .30902 0 4.5 ] [.01131 .41202 -6 -4.5 ] [.01131 .41202 0 4.5 ] [.01131 .51503 -6 -4.5 ] [.01131 .51503 0 4.5 ] [.01131 .61803 -6 -4.5 ] [.01131 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .15079 .10301 m .15079 .10926 L s [(2)] .15079 .09051 0 1 Mshowa .27778 .10301 m .27778 .10926 L s [(4)] .27778 .09051 0 1 Mshowa .40476 .10301 m .40476 .10926 L s [(6)] .40476 .09051 0 1 Mshowa .53175 .10301 m .53175 .10926 L s [(8)] .53175 .09051 0 1 Mshowa .65873 .10301 m .65873 .10926 L s [(10)] .65873 .09051 0 1 Mshowa .78571 .10301 m .78571 .10926 L s [(12)] .78571 .09051 0 1 Mshowa .9127 .10301 m .9127 .10926 L s [(14)] .9127 .09051 0 1 Mshowa .125 Mabswid .05556 .10301 m .05556 .10676 L s .0873 .10301 m .0873 .10676 L s .11905 .10301 m .11905 .10676 L s .18254 .10301 m .18254 .10676 L s .21429 .10301 m .21429 .10676 L s .24603 .10301 m .24603 .10676 L s .30952 .10301 m .30952 .10676 L s .34127 .10301 m .34127 .10676 L s .37302 .10301 m .37302 .10676 L s .43651 .10301 m .43651 .10676 L s .46825 .10301 m .46825 .10676 L s .5 .10301 m .5 .10676 L s .56349 .10301 m .56349 .10676 L s .59524 .10301 m .59524 .10676 L s .62698 .10301 m .62698 .10676 L s .69048 .10301 m .69048 .10676 L s .72222 .10301 m .72222 .10676 L s .75397 .10301 m .75397 .10676 L s .81746 .10301 m .81746 .10676 L s .84921 .10301 m .84921 .10676 L s .88095 .10301 m .88095 .10676 L s .94444 .10301 m .94444 .10676 L s .97619 .10301 m .97619 .10676 L s .25 Mabswid 0 .10301 m 1 .10301 L s .02381 0 m .03006 0 L s [(-1)] .01131 0 1 0 Mshowa .02381 .20601 m .03006 .20601 L s [(1)] .01131 .20601 1 0 Mshowa .02381 .30902 m .03006 .30902 L s [(2)] .01131 .30902 1 0 Mshowa .02381 .41202 m .03006 .41202 L s [(3)] .01131 .41202 1 0 Mshowa .02381 .51503 m .03006 .51503 L s [(4)] .01131 .51503 1 0 Mshowa .02381 .61803 m .03006 .61803 L s [(5)] .01131 .61803 1 0 Mshowa .125 Mabswid .02381 .0206 m .02756 .0206 L s .02381 .0412 m .02756 .0412 L s .02381 .0618 m .02756 .0618 L s .02381 .0824 m .02756 .0824 L s .02381 .12361 m .02756 .12361 L s .02381 .14421 m .02756 .14421 L s .02381 .16481 m .02756 .16481 L s .02381 .18541 m .02756 .18541 L s .02381 .22661 m .02756 .22661 L s .02381 .24721 m .02756 .24721 L s .02381 .26781 m .02756 .26781 L s .02381 .28842 m .02756 .28842 L s .02381 .32962 m .02756 .32962 L s .02381 .35022 m .02756 .35022 L s .02381 .37082 m .02756 .37082 L s .02381 .39142 m .02756 .39142 L s .02381 .43262 m .02756 .43262 L s .02381 .45322 m .02756 .45322 L s .02381 .47383 m .02756 .47383 L s .02381 .49443 m .02756 .49443 L s .02381 .53563 m .02756 .53563 L s .02381 .55623 m .02756 .55623 L s .02381 .57683 m .02756 .57683 L s .02381 .59743 m .02756 .59743 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .10301 m .02499 .10301 L .02605 .10301 L .02729 .10301 L .02846 .10301 L .02954 .10301 L .03053 .10301 L .03163 .10301 L .03279 .10301 L .03395 .10301 L .0352 .10301 L .03638 .10301 L .03746 .10301 L .03873 .10301 L .04011 .10301 L .04141 .10301 L .04262 .10301 L .04386 .10301 L .04521 .10302 L .0464 .10302 L .04765 .10302 L .05003 .10303 L .05221 .10304 L .05474 .10306 L .05709 .10308 L .05968 .10312 L .06244 .10316 L .06526 .10322 L .06787 .10329 L .07067 .10338 L .07376 .1035 L .07896 .10378 L .08448 .10417 L .08947 .10463 L .09485 .10526 L .10458 .10675 L .11415 .1087 L .12462 .11132 L .14572 .11753 L .15581 .12043 L .16533 .12274 L .1706 .12375 L .17337 .12417 L .17631 .12454 L .17878 .12478 L .18018 .12488 L .1815 .12496 L .1827 .12501 L .1838 .12505 L .18507 .12507 L Mistroke .18628 .12507 L .18753 .12505 L .18823 .12503 L .18888 .125 L .19007 .12494 L .19132 .12486 L .1935 .12466 L .19589 .12438 L .19845 .12398 L .20119 .12346 L .20613 .12225 L .21576 .11898 L .22453 .11506 L .24526 .10361 L .25438 .09845 L .26442 .09337 L .26905 .09139 L .27403 .0896 L .27874 .08828 L .28082 .08782 L .28305 .08742 L .2843 .08724 L .28543 .0871 L .28659 .08698 L .28765 .0869 L .28889 .08683 L .28955 .08681 L .29024 .0868 L .2914 .0868 L .29268 .08683 L .294 .0869 L .2952 .087 L .29659 .08715 L .29791 .08733 L .30026 .08774 L .30281 .08832 L .30764 .08981 L .31219 .09163 L .32256 .09718 L .34355 .11236 L .36201 .12578 L .36706 .12878 L .37247 .13146 L .37509 .13254 L .37757 .1334 L .37978 .13405 L .38221 .13462 L .38338 .13483 L .38463 .13502 L .38533 .13511 L Mistroke .38596 .13518 L .3872 .13528 L .38833 .13533 L .38957 .13535 L .39073 .13532 L .39181 .13527 L .39247 .13522 L .39317 .13515 L .39446 .13498 L .3957 .13478 L .39685 .13455 L .39947 .13388 L .40223 .13297 L .40743 .13065 L .41228 .12782 L .42332 .11929 L .44197 .10049 L .46225 .08001 L .4676 .07564 L .47329 .07178 L .47814 .06922 L .48091 .06811 L .4835 .0673 L .48596 .06674 L .4872 .06655 L .4879 .06646 L .48856 .0664 L .48973 .06632 L .4908 .0663 L .49203 .06632 L .49319 .0664 L .49446 .06654 L .49509 .06663 L .4958 .06675 L .4982 .0673 L .49955 .06771 L .501 .06822 L .50363 .06934 L .50889 .07232 L .51377 .07592 L .52482 .08665 L .54492 .11173 L .56377 .13434 L .56894 .13924 L .57438 .14347 L .57903 .14623 L .58167 .14742 L .58413 .14826 L .58546 .14861 L .58617 .14876 L Mistroke .58692 .1489 L .58757 .149 L .58829 .14909 L .58955 .14919 L .59078 .14921 L .59211 .14917 L .59335 .14905 L .5945 .14888 L .59591 .14859 L .59722 .14824 L .60018 .14717 L .6028 .14589 L .60559 .1442 L .61104 .13995 L .61618 .13489 L .62578 .12307 L .64415 .09539 L .65458 .07947 L .66432 .06641 L .66926 .06089 L .67394 .05652 L .67652 .0545 L .67933 .05266 L .68187 .05133 L .68425 .05036 L .68546 .04998 L .68677 .04966 L .68747 .04952 L .68822 .0494 L .68886 .04932 L .68955 .04926 L .69029 .04923 L .69096 .04922 L .69174 .04924 L .69247 .04929 L .69377 .04945 L .69451 .04958 L .69519 .04973 L .69753 .05042 L .69886 .05094 L .7001 .05152 L .70531 .05479 L .71028 .05918 L .71571 .06527 L .72559 .0793 L .74412 .11172 L .75512 .13112 L .76547 .14687 L .76999 .15253 L .77476 .15747 L Mistroke .77726 .15961 L .78 .16156 L .78249 .16297 L .78481 .16396 L .78607 .16437 L .78724 .16466 L .78853 .16488 L .78918 .16496 L .78989 .16501 L .79109 .16503 L .79238 .16495 L .79359 .16478 L .7947 .16455 L .79579 .16425 L .7968 .16391 L .7991 .16289 L .80133 .1616 L .80372 .15989 L .80918 .15475 L .81434 .14842 L .82398 .1333 L .84232 .09711 L .85274 .07589 L .86248 .05816 L .87208 .04441 L .87485 .04133 L .87747 .03883 L .87981 .03696 L .88239 .03529 L .88492 .03408 L .88631 .0336 L .88762 .03327 L .88883 .03306 L .88995 .03296 L .89117 .03296 L .89247 .03306 L .89378 .03328 L .89451 .03346 L .89518 .03365 L .8977 .03467 L .89912 .03543 L .90065 .0364 L .90344 .03859 L .90832 .04362 L .91288 .04966 L .92328 .06754 L .94431 .11425 L .97619 .17342 L Mfstroke .02381 .20601 m .02499 .20601 L .02605 .20601 L .02729 .20601 L .02846 .20601 L .02954 .20601 L .03053 .20601 L .03163 .20601 L .03279 .206 L .03395 .206 L .0352 .206 L .03638 .20599 L .03746 .20598 L .03873 .20597 L .04011 .20595 L .04262 .2059 L .04396 .20587 L .04523 .20583 L .04759 .20574 L .05014 .20562 L .0529 .20544 L .05532 .20524 L .05787 .20499 L .06244 .2044 L .06746 .20354 L .07283 .20231 L .08253 .19924 L .09406 .19407 L .10458 .18795 L .14443 .15753 L .16264 .14391 L .17302 .13727 L .18275 .13197 L .19307 .12741 L .20253 .12414 L .21264 .12149 L .22354 .11943 L .23334 .11811 L .24246 .11718 L .26316 .11562 L .27368 .11499 L .28362 .11449 L .28795 .11431 L .29254 .11415 L .29493 .11409 L .29757 .11403 L .29872 .11401 L .29996 .11399 L .30112 .11397 L .30221 .11396 L Mistroke .30285 .11396 L .30354 .11395 L .30475 .11395 L .30544 .11395 L .30617 .11395 L .30746 .11395 L .30867 .11396 L .30979 .11397 L .31102 .11398 L .31234 .114 L .31531 .11407 L .31812 .11415 L .32338 .11437 L .32839 .11463 L .33302 .11493 L .34348 .11573 L .35249 .11645 L .35722 .1168 L .36218 .11711 L .36484 .11725 L .36774 .11737 L .36901 .11741 L .37038 .11744 L .37166 .11747 L .37283 .11748 L .37411 .11749 L .37529 .11749 L .37659 .11748 L .37724 .11747 L .37796 .11746 L .37917 .11743 L .38046 .11739 L .3828 .11728 L .38536 .11712 L .38808 .1169 L .39296 .11634 L .39821 .11552 L .40402 .11434 L .41439 .11151 L .42413 .10809 L .44258 .10035 L .45304 .09587 L .46279 .09216 L .46776 .09057 L .47246 .0893 L .47506 .08872 L .47788 .08819 L .48044 .08781 L .4828 .08755 L .48401 .08744 L Mistroke .48532 .08736 L .48655 .08731 L .48768 .08728 L .48899 .08727 L .48974 .08728 L .49043 .0873 L .49169 .08735 L .4924 .08739 L .49304 .08743 L .49573 .0877 L .49724 .0879 L .49863 .08812 L .50389 .08926 L .5086 .09066 L .51364 .09257 L .52273 .09691 L .5434 .10964 L .55302 .11581 L .56323 .12164 L .56855 .12416 L .57443 .12639 L .577 .12716 L .57977 .12783 L .58104 .12808 L .58238 .12831 L .58353 .12847 L .58478 .12861 L .58607 .12872 L .58728 .12878 L .58846 .12881 L .58955 .1288 L .59073 .12877 L .59203 .12869 L .59268 .12863 L .59338 .12856 L .59465 .1284 L .59711 .12797 L .59945 .12742 L .60383 .12604 L .609 .12381 L .61379 .1212 L .62465 .11361 L .6435 .09702 L .6534 .08809 L .66385 .07978 L .66919 .07628 L .67504 .07321 L .67761 .07215 L .68036 .07123 L .68169 .07086 L Mistroke .68295 .07056 L .68417 .07031 L .68532 .07013 L .68648 .06998 L .68776 .06987 L .68901 .06981 L .68971 .0698 L .69035 .06981 L .69149 .06985 L .69273 .06994 L .6939 .07008 L .69499 .07025 L .69766 .07082 L .70008 .07155 L .70287 .07262 L .7055 .07386 L .71526 .08026 L .72557 .08969 L .74404 .11051 L .75423 .12212 L .76389 .13179 L .77257 .13857 L .77747 .14136 L .7798 .1424 L .782 .14319 L .78416 .14378 L .78538 .14404 L .78649 .14422 L .78765 .14436 L .78891 .14445 L .79023 .14447 L .79145 .14443 L .79276 .14431 L .794 .14414 L .79511 .14394 L .79631 .14366 L .79901 .14281 L .8015 .14177 L .80634 .13904 L .81166 .13502 L .82241 .12403 L .84348 .0956 L .86274 .07021 L .87226 .06096 L .87487 .059 L .87765 .05722 L .88025 .05585 L .88265 .05487 L .8851 .05413 L .88639 .05386 L Mistroke .88704 .05375 L .88776 .05366 L .88896 .05356 L .89025 .05353 L .89146 .05357 L .89257 .05367 L .89372 .05384 L .89478 .05406 L .89718 .05475 L .8993 .05559 L .90155 .05672 L .90645 .06001 L .91094 .06397 L .92105 .07582 L .93934 .10428 L .97619 .15523 L Mfstroke .02381 .30902 m .02499 .30902 L .02605 .30902 L .02729 .30902 L .02846 .30902 L .02954 .30901 L .03053 .30901 L .03163 .30901 L .03279 .309 L .03395 .309 L .0352 .30898 L .03638 .30897 L .03746 .30895 L .03873 .30892 L .04011 .30889 L .04262 .30879 L .04396 .30872 L .04523 .30864 L .04759 .30846 L .05014 .3082 L .0529 .30783 L .05532 .30742 L .05787 .30689 L .06244 .30565 L .06746 .30379 L .07283 .30116 L .08253 .29447 L .08845 .28906 L .09406 .28298 L .10458 .26914 L .14291 .20079 L .16229 .16623 L .17259 .15098 L .178 .14417 L .18369 .13798 L .18874 .13339 L .19426 .12933 L .19928 .12651 L .20394 .12461 L .20642 .12387 L .20876 .12335 L .21086 .12301 L .21205 .12287 L .21314 .12277 L .21431 .12271 L .21541 .12268 L .21661 .12268 L .21791 .12272 L .21856 .12275 L .21926 .1228 L Mistroke .22055 .12292 L .2218 .12307 L .22296 .12324 L .22805 .12427 L .23355 .12585 L .24352 .12955 L .26214 .13694 L .26674 .13846 L .27167 .13981 L .2745 .14044 L .27708 .14092 L .27967 .14129 L .28212 .14154 L .28333 .14163 L .28403 .14167 L .28467 .1417 L .28534 .14172 L .28606 .14174 L .28737 .14175 L .2885 .14173 L .28973 .14168 L .29102 .14161 L .29222 .14151 L .29365 .14136 L .295 .14119 L .29753 .14079 L .30025 .14023 L .3032 .13948 L .31268 .13617 L .32311 .13118 L .34417 .11873 L .35501 .11226 L .36529 .10676 L .37529 .10237 L .37969 .1008 L .38456 .09934 L .3889 .09829 L .39358 .09742 L .39616 .09705 L .39852 .09679 L .40079 .0966 L .40193 .09653 L .40317 .09646 L .40383 .09644 L .40454 .09641 L .40583 .09638 L .40708 .09637 L .40824 .09638 L .4095 .0964 L .41021 .09641 L Mistroke .41087 .09643 L .41222 .09649 L .41366 .09657 L .41628 .09675 L .41877 .09698 L .42343 .09753 L .43303 .09905 L .44326 .101 L .46165 .10457 L .47194 .10625 L .47736 .10698 L .48307 .10763 L .48809 .1081 L .49358 .10852 L .49858 .10881 L .50325 .10902 L .50826 .10919 L .51109 .10926 L .51371 .10932 L .51852 .10939 L .52094 .10942 L .52362 .10944 L .52823 .10947 L .53319 .1095 L .53754 .10952 L .54218 .10953 L .54478 .10954 L .54714 .10954 L .54828 .10955 L .54952 .10955 L .55068 .10955 L .55176 .10955 L .55301 .10955 L .55418 .10955 L .55547 .10955 L .55619 .10954 L .55685 .10954 L .55817 .10954 L .55961 .10953 L .56097 .10952 L .56223 .10951 L .565 .10948 L .5676 .10944 L .56994 .1094 L .57246 .10933 L .57522 .10925 L .57813 .10914 L .58334 .10887 L .58832 .10852 L .59285 .10812 L Mistroke .60303 .10688 L .61246 .10529 L .62138 .1034 L .63965 .09869 L .65004 .09586 L .65978 .09345 L .66468 .09242 L .66933 .09159 L .6719 .0912 L .6747 .09085 L .67725 .09059 L .67961 .09041 L .68093 .09034 L .68162 .09031 L .68236 .09028 L .683 .09026 L .68371 .09025 L .68495 .09023 L .68615 .09024 L .68745 .09026 L .68867 .0903 L .68979 .09036 L .69117 .09044 L .69245 .09055 L .69536 .09086 L .69792 .09122 L .70064 .09169 L .70552 .09278 L .71 .09404 L .7201 .0977 L .73838 .10647 L .74829 .11164 L .75889 .11678 L .76404 .11894 L .7689 .1207 L .77322 .12199 L .77569 .12259 L .77799 .12306 L .78021 .12343 L .78261 .12372 L .78392 .12383 L .78516 .12391 L .78628 .12396 L .7875 .12398 L .78865 .12397 L .7897 .12394 L .79091 .12388 L .79205 .12379 L .7931 .12369 L .79407 .12357 L Mistroke .79626 .12324 L .79864 .12276 L .80086 .12222 L .80591 .1206 L .81082 .11855 L .81608 .11587 L .83592 .10237 L .84638 .09425 L .85748 .08608 L .86734 .08006 L .87295 .07743 L .87552 .07645 L .87821 .0756 L .88051 .07501 L .88173 .07476 L .88303 .07453 L .88422 .07436 L .88531 .07424 L .88641 .07415 L .88744 .07409 L .88864 .07407 L .88992 .07409 L .89101 .07414 L .89221 .07423 L .89349 .07438 L .89487 .0746 L .89737 .07512 L .89992 .07584 L .90264 .07681 L .90752 .07905 L .91277 .08216 L .91858 .08636 L .93869 .10522 L .97619 .13704 L Mfstroke .02381 .41202 m .02499 .41202 L .02605 .41202 L .02729 .41202 L .02846 .41202 L .02954 .41202 L .03053 .41202 L .03163 .41201 L .03279 .412 L .03395 .41199 L .0352 .41197 L .03638 .41195 L .03746 .41192 L .03873 .41188 L .04011 .41183 L .04262 .41168 L .04396 .41157 L .04523 .41146 L .04759 .41118 L .05014 .41078 L .0529 .41022 L .05532 .4096 L .05787 .40879 L .06244 .40689 L .06746 .40405 L .07283 .40001 L .08253 .38969 L .08845 .38132 L .09406 .37188 L .10458 .35034 L .14291 .24284 L .16229 .18831 L .17259 .16444 L .178 .15389 L .18369 .14445 L .18874 .13758 L .19426 .1317 L .19668 .12966 L .19928 .12785 L .2017 .12648 L .20394 .1255 L .20512 .1251 L .2064 .12473 L .2077 .12445 L .20841 .12433 L .20907 .12425 L .21029 .12414 L .2114 .12411 L .21269 .12414 L .21391 .12424 L Mistroke .21499 .12439 L .21615 .1246 L .21824 .12513 L .22071 .12598 L .22296 .12695 L .22785 .12964 L .23301 .13321 L .24224 .14087 L .2523 .14994 L .26308 .15891 L .26837 .16257 L .27338 .1654 L .27787 .16731 L .28041 .16811 L .28154 .1684 L .28275 .16865 L .28389 .16885 L .28494 .16899 L .28594 .16908 L .28699 .16915 L .28814 .16917 L .28922 .16915 L .2904 .16907 L .29166 .16894 L .29288 .16875 L .2942 .16849 L .29661 .16784 L .29895 .16702 L .3011 .16608 L .30579 .16348 L .31082 .15987 L .3199 .15147 L .34056 .1264 L .36034 .10159 L .37153 .09032 L .37687 .08609 L .38188 .08289 L .38629 .08073 L .38878 .0798 L .39105 .07913 L .39341 .07861 L .39474 .0784 L .39598 .07825 L .3973 .07815 L .3985 .07811 L .39916 .07811 L .39988 .07812 L .40118 .07819 L .40245 .07831 L .40365 .07846 L Mistroke .40496 .07868 L .40636 .07897 L .40923 .07975 L .41185 .08066 L .4169 .08292 L .4217 .08561 L .45991 .11534 L .47005 .12201 L .47502 .12462 L .4796 .12657 L .48212 .12745 L .48489 .12824 L .48767 .12886 L .48919 .12912 L .4906 .12931 L .4919 .12944 L .49312 .12954 L .49422 .12959 L .49542 .12961 L .49672 .1296 L .49737 .12958 L .4981 .12954 L .49938 .12945 L .50058 .12934 L .50301 .129 L .50528 .12856 L .51042 .12719 L .51543 .12536 L .52077 .12294 L .53928 .1121 L .55921 .09959 L .5697 .09416 L .57544 .09178 L .5808 .08999 L .58558 .08877 L .58827 .08825 L .59072 .08788 L .59201 .08772 L .59339 .08758 L .59419 .08751 L .59491 .08746 L .59564 .08742 L .59631 .08739 L .59763 .08734 L .59832 .08733 L .59905 .08733 L .60027 .08734 L .60097 .08735 L .60162 .08738 L .60281 .08743 L Mistroke .60411 .08752 L .60647 .08773 L .60877 .08801 L .61089 .08833 L .6157 .08926 L .6208 .09053 L .65863 .1037 L .66857 .10677 L .67803 .10905 L .68324 .11002 L .68599 .11043 L .68893 .11081 L .69164 .1111 L .6941 .11131 L .69657 .11147 L .6989 .11159 L .70006 .11163 L .70131 .11166 L .70248 .11168 L .70355 .11169 L .70481 .1117 L .70552 .1117 L .70617 .11169 L .70738 .11167 L .70866 .11165 L .70987 .11161 L .71119 .11157 L .71356 .11146 L .71897 .11111 L .72366 .11071 L .7286 .11021 L .73744 .10918 L .77554 .10475 L .81608 .10082 L .83481 .09854 L .85512 .09594 L .86051 .09536 L .86622 .09486 L .86857 .09469 L .87109 .09453 L .87254 .09446 L .87386 .09441 L .87512 .09436 L .87645 .09433 L .87715 .09431 L .8778 .0943 L .87902 .09429 L .88016 .09429 L .8814 .09429 L .88205 .0943 L Mistroke .88276 .09431 L .88351 .09433 L .88421 .09435 L .88545 .09439 L .88679 .09445 L .88943 .09461 L .89192 .09481 L .8966 .09532 L .9012 .096 L .90628 .09696 L .91654 .09954 L .93497 .10584 L .94521 .10977 L .95598 .11371 L .96517 .11655 L .97038 .11781 L .97521 .11871 L .97619 .11885 L Mfstroke .02381 .51503 m .02499 .51503 L .02605 .51503 L .02729 .51503 L .02846 .51503 L .02954 .51502 L .03053 .51502 L .03163 .51501 L .03279 .515 L .03395 .51499 L .0352 .51496 L .03638 .51493 L .03746 .5149 L .03873 .51484 L .04011 .51476 L .04262 .51457 L .04396 .51443 L .04523 .51427 L .04759 .5139 L .05014 .51337 L .0529 .51261 L .05532 .51177 L .05787 .51068 L .06244 .50813 L .06746 .50431 L .07283 .49885 L .08253 .48492 L .08845 .47358 L .09406 .46079 L .10458 .43154 L .14291 .28489 L .16229 .21039 L .17259 .17789 L .178 .16362 L .18369 .15092 L .18874 .14177 L .19141 .13774 L .19426 .13407 L .19689 .13126 L .19928 .12918 L .20168 .12754 L .20394 .12639 L .20504 .12598 L .20625 .12562 L .20751 .12536 L .20869 .12523 L .20979 .12519 L .2108 .12522 L .21196 .12535 L .21307 .12555 L Mistroke .21426 .12586 L .21552 .12628 L .2178 .12729 L .22051 .12888 L .22296 .13066 L .23301 .14074 L .24224 .15271 L .25302 .1677 L .26308 .18055 L .27219 .18965 L .27481 .19165 L .27729 .19326 L .2795 .19443 L .28194 .19545 L .28323 .19586 L .28444 .19616 L .28572 .1964 L .28643 .19649 L .28709 .19655 L .2882 .1966 L .28941 .19658 L .29067 .19647 L .29185 .19628 L .29252 .19614 L .29326 .19596 L .29458 .19555 L .29706 .19453 L .29995 .19291 L .30262 .191 L .30731 .18674 L .31234 .18094 L .32142 .16765 L .34207 .12882 L .36186 .09131 L .36728 .08271 L .37304 .07482 L .37803 .06919 L .38064 .06672 L .3834 .06449 L .38583 .06285 L .38849 .06142 L .38965 .06091 L .3909 .06045 L .39207 .06009 L .39315 .05982 L .39434 .05961 L .39562 .05945 L .39675 .05939 L .39795 .0594 L .39903 .05948 L Mistroke .40003 .0596 L .40113 .0598 L .40232 .06008 L .40499 .06097 L .40791 .06235 L .41323 .06589 L .41841 .07048 L .42321 .07562 L .44202 .10109 L .46236 .12949 L .46779 .13573 L .47353 .14131 L .47842 .14512 L .48123 .14687 L .48379 .14819 L .48638 .14923 L .48883 .14995 L .49017 .15024 L .49141 .15043 L .49206 .1505 L .49278 .15056 L .49353 .1506 L .49423 .15061 L .49542 .15059 L .49667 .15049 L .49774 .15036 L .49891 .15015 L .50018 .14986 L .50153 .14948 L .50397 .1486 L .50656 .1474 L .5093 .14584 L .51422 .14236 L .52533 .13169 L .54548 .10662 L .55565 .09381 L .5653 .08309 L .57396 .0754 L .57885 .07207 L .58339 .0697 L .58594 .06869 L .58834 .06795 L .59049 .06747 L .59171 .06728 L .59282 .06714 L .59403 .06705 L .59515 .06701 L .59638 .06702 L .5977 .06709 L .59837 .06715 L Mistroke .59909 .06723 L .6004 .06743 L .60167 .06768 L .60285 .06796 L .60768 .06959 L .61042 .07084 L .61299 .07223 L .62375 .07989 L .64228 .09761 L .66251 .11708 L .66781 .12127 L .67346 .12507 L .67835 .12774 L .68363 .12991 L .68646 .13076 L .6877 .13106 L .68906 .13133 L .69026 .13154 L .69154 .13171 L .69225 .13178 L .6929 .13184 L .69417 .13191 L .69539 .13194 L .69652 .13193 L .69776 .13188 L .69907 .13178 L .70021 .13165 L .70128 .1315 L .70372 .13105 L .70615 .13044 L .70841 .12974 L .71355 .12772 L .72389 .12202 L .7424 .10856 L .76112 .09482 L .76625 .09164 L .77169 .08871 L .77638 .08659 L .78143 .08477 L .78417 .08399 L .78668 .08341 L .7892 .08296 L .79159 .08265 L .79293 .08253 L .79368 .08247 L .79436 .08243 L .795 .08241 L .79569 .08239 L .79693 .08238 L .79764 .08238 L Mistroke .79841 .0824 L .79922 .08243 L .79997 .08247 L .80132 .08258 L .80277 .08272 L .80536 .08308 L .80818 .08361 L .81331 .0849 L .82291 .08833 L .84233 .0977 L .85264 .10295 L .86374 .10804 L .8689 .11006 L .87432 .11189 L .87935 .11328 L .88402 .1143 L .88842 .11502 L .89088 .11531 L .89311 .11551 L .89439 .1156 L .89556 .11566 L .89686 .11571 L .89751 .11573 L .89822 .11575 L .89942 .11576 L .90071 .11575 L .90193 .11573 L .90304 .11569 L .90422 .11564 L .90532 .11557 L .90784 .11538 L .91029 .11514 L .91293 .11481 L .9221 .11325 L .93291 .11078 L .94284 .10821 L .97619 .10066 L Mfstroke .02381 .61803 m .02499 .61803 L .02605 .61803 L .02729 .61803 L .02846 .61803 L .02954 .61803 L .03053 .61802 L .03163 .61802 L .03279 .618 L .03395 .61798 L .0352 .61795 L .03638 .61791 L .03746 .61787 L .03873 .6178 L .04011 .6177 L .04262 .61746 L .04396 .61728 L .04523 .61709 L .04759 .61662 L .05014 .61595 L .0529 .615 L .05532 .61395 L .05787 .61258 L .06244 .60937 L .06746 .60457 L .07283 .5977 L .08253 .58014 L .08845 .56585 L .09406 .5497 L .10458 .51273 L .14291 .32695 L .16229 .23247 L .17259 .19135 L .178 .17335 L .18369 .15739 L .18874 .14596 L .19141 .14097 L .19426 .13645 L .19689 .13302 L .19928 .13052 L .20168 .12859 L .20394 .12729 L .20504 .12684 L .20625 .12647 L .20751 .12624 L .20869 .12616 L .20979 .1262 L .2108 .12632 L .21196 .12658 L .21307 .12694 L Mistroke .21426 .12743 L .21552 .12809 L .2178 .12958 L .22296 .13437 L .22833 .14116 L .23327 .1487 L .24441 .16861 L .25485 .18819 L .26459 .20451 L .26927 .21102 L .2737 .21613 L .27615 .21845 L .27881 .22055 L .28122 .22204 L .28345 .22306 L .28464 .22346 L .28592 .22377 L .28712 .22396 L .28823 .22404 L .28945 .22402 L .29015 .22395 L .29079 .22387 L .2921 .22359 L .29283 .22338 L .29349 .22315 L .29612 .22193 L .2976 .22101 L .29897 .22001 L .30413 .21502 L .30882 .20884 L .31385 .2006 L .32293 .18206 L .34359 .12913 L .36337 .07934 L .3688 .0682 L .37456 .05814 L .37948 .05122 L .38213 .04819 L .38492 .04552 L .38737 .04364 L .38867 .04283 L .39007 .0421 L .39135 .04155 L .39251 .04116 L .39368 .04087 L .39479 .04069 L .39608 .0406 L .39673 .0406 L .39745 .04063 L .39871 .04079 L Mistroke .3999 .04104 L .40061 .04124 L .40129 .04146 L .40257 .04197 L .40544 .04354 L .40776 .04522 L .41023 .04739 L .41467 .05226 L .42473 .06724 L .44472 .10693 L .45523 .12857 L .46633 .14852 L .47114 .15562 L .47629 .16194 L .47925 .1649 L .48194 .16714 L .48468 .16897 L .48589 .16963 L .48721 .17025 L .48832 .17068 L .48951 .17105 L .49076 .17135 L .49194 .17153 L .49325 .17164 L .4939 .17165 L .49462 .17163 L .49589 .17151 L .49708 .1713 L .4983 .171 L .49945 .17063 L .50159 .16972 L .50391 .16841 L .50642 .16664 L .51131 .16215 L .51589 .15679 L .52631 .14116 L .54736 .10153 L .56595 .06893 L .57104 .06194 L .57646 .0558 L .57909 .05334 L .58157 .05136 L .58378 .04987 L .5862 .04856 L .58879 .04752 L .5902 .04711 L .59151 .04683 L .59274 .04666 L .59387 .04658 L .59516 .04657 L Mistroke .59639 .04666 L .59702 .04673 L .59772 .04684 L .59913 .04715 L .60044 .04753 L .60166 .04797 L .60442 .04927 L .60739 .0511 L .61277 .05554 L .61789 .06096 L .6275 .07375 L .64752 .10615 L .66568 .13336 L .67054 .1391 L .6758 .14427 L .68077 .14807 L .68298 .14939 L .68534 .15056 L .68666 .15109 L .68785 .1515 L .68907 .15185 L .6902 .1521 L .69151 .15232 L .6922 .15241 L .69294 .15247 L .69417 .15251 L .69488 .1525 L .69554 .15248 L .69685 .15235 L .69752 .15226 L .69824 .15214 L .69952 .15186 L .70072 .15152 L .70342 .15053 L .70631 .1491 L .71164 .14553 L .71655 .14126 L .72765 .12877 L .7478 .10091 L .75684 .08867 L .76655 .07726 L .77212 .0719 L .77721 .06792 L .77966 .06635 L .78233 .0649 L .78486 .06378 L .78719 .06297 L .78841 .06264 L .78975 .06234 L .79109 .06212 L Mistroke .79183 .06203 L .79251 .06197 L .79378 .0619 L .79493 .06189 L .79557 .06191 L .79626 .06195 L .79752 .06207 L .79865 .06223 L .79989 .06247 L .80239 .06313 L .80471 .06395 L .80689 .06491 L .81216 .06791 L .81699 .07143 L .82787 .08157 L .84843 .10492 L .85844 .11587 L .86754 .12435 L .8721 .12787 L .87703 .13104 L .8817 .13336 L .88377 .13417 L .88599 .13489 L .88722 .13522 L .88835 .13548 L .88949 .13569 L .89055 .13586 L .89178 .136 L .89243 .13606 L .89312 .1361 L .89428 .13614 L .89556 .13613 L .89687 .13607 L .89807 .13597 L .89939 .1358 L .90078 .13556 L .90312 .13503 L .90568 .13426 L .91095 .13209 L .91594 .12937 L .92526 .12286 L .9462 .1045 L .97619 .08247 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[oooclo5H/1EGooj[ooc/k>gYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00000EED0EJX00=d81Slo?ginO[fm_Oclo0<30d92@X61PL30`000 EED0EJX0EOl0E@00ZUD0ZZX0E@<30d92@X61PL30`000E@L71dI6AXF5QLC4a000ZUD0ZZX0Z_l0ZP00 e5D0ojX0ZPL71dI6AXF5QLC4a000ZP/;2dY:BXV9RLS8b000oeD0ojX0ool0o`1E05EE0:X0o`/;2dY: BXV9RLS8b000o`l?3di>CXf=SLcCXf=SLcH000?o00000P000ol0000M00004P02of801?mo00;o1P000ol0000L00004P000ol0001O00?o 0`000ol0001l0003o`0000H000?o00006`00018000?o0000GP000ol000050003o`0002d01_lF00;o <0000ol000070003o`0001X0000B0003o`0005h000?o00001P000ol0000[0003o`0000@000?o0000 4003o`040?ooobd000?o00002@000ol0000I00004P000ol0001M0003o`0000L000?o0000:@02o`P0 00?o00003P000ol0000500;o:P001?l00007o`8000?o00006@00018000?o0000G0000ol0000200Co 0`000ol0000X0003o`0000L000?o00003@000ol000080003o`0002H000?o00000_l70004o`00oaX0 000B00;oG@000ol00002o`@00_l20003o`0002H000?o00002@000ol0000;0003o`0000T000?o0000 4@06o`l000Co0?oo2@000ol0o`0J00004P000ol0000N00Co7`03oaL000?o00000_l400;o00D0o`00 o`0?00Go1`02o`T000?o00000`04o`@000?o00000`03o`<00ol;0003o`0000/01Ol00`3oo`02o`03 0?l000`000?o0?l01`03o`040?l000?o5`00018000?o00007`000ol0000P0003o`0001H000?o0?l0 1@002?l00?l0o`3o3@02o`D00_l40004o`00o`L000?o00000P02o`<00ol30003o`0000<000?o0000 0P000ol00007o`@000?o00003003o`<000?o00000P03o`T000?o0?l02@001_l0003o00;o60000180 00?o000080000ol0000M00Co5`02o`H000Co003o0P000ol0o`0;0003o`0000L000Go0000o`020003 o`0000D000Go0000o`0800;o0P000ol000020004o`0000;o00@0oooo1@03o`8000?o00002`000ol0 00040003o`0000@000?o00001P000ol0o`0400Ko1004oaL0000B0003o`00024000?o000070000ol0 o`0G0003o`3o00D01?l400;o2@02o`/000Co00000_l70003o`0000;o2`000ol0o`040005o`00o`00 0_l00`3o00080004o`00o`/00_l70003o`0000@000?o000010000ol0o`0200?o1@02o`<00olH0000 4P02oal000Co003o7P000ol0o`0F0003o`3o00<00ol00`3o0003o`8000?o0?l01`000ol0000<0003 o`3o008000?o000010000ol0o`0>0003o`3o00<000Co0?l00_l20003o`0000P00_l00`3o00080003 o`3o00@01?l70003o`0000<00_l00`3oo`080003o`3o00Ko2`02o`X0000B0003o`0001l00_lP00;o 5P02o`800_l500?o00<0ool00ol60003o`0000h000Co0?oo1P000ol0o`0@0004o`3o00;o00<0ool0 0P02o`d00_l80004o`3oo`802ol20003o`0000800_l00`3o000800;o0`000ol0o`02o`X000?o0000 2P00018000?o0000@@0:o`d000?o0?l00P000ol0000;00?o10000ol0000200[o10000ol000050004 o`00o`@02_l300;o0`02oa<00_l60004o`0000?o2`02o`060?l00?l00_l@00;o00<0ool020000ol0 000;00004P000ol0000n00?o2P03o`X00_l20003o`0000d00_l30003o`3o00?o2P04o`030?l000D0 00?o0?l00P03o`X00ol0103oo`02o`030?oo00go10000ol0o`050004o`3ooa000ol0103ooolC0003 o`3o00H00_l>00004P000ol0000m0003o`0000h00_l800;o00<0o`001@08o`<00_l0103ooolB00?o 10001?l0o`02oa001ol?00;o0`02o`@00olD01Ko0`02o`D000?o00003P00018000?o0000>`02oa<0 00?o00001002o`030?l000801Ol700Oo5P02o`<01?lB00Oo4@02o`800_l200;o3@0=oa800_l200;o 0`000ol0000200?o2P0000`0oolAo`@0000B0003o`0000H000?o00001@000ol0o`05o`030?l000L0 00?o00001@000ol000060003o`0000800_l00`3o00070003o`0000H000?o00001002o`801ol80004 o`0000;o0P000ol0o`020004o`0000Oo00<0o`001`000ol0o`03o`<000?o00001007o`<00_l0103o 0002o`030?l000?o0`000ol000060004o`0000?o00<0ool02?l80006o`00o`3o0P001?l00002o`<0 00?o00001P000ol0000300Co1`000ol0000800004P000ol0000F00Go1@000ol0000B0003o`0000X0 00?o00003P000ol00003o``01?l30003o`0000;o0P000ol0000701[o2P001?l00?l20004o`3oo`@0 5Ol80003o`0000L00_l00`3o00030003o`0000800ol90004o`0000Oo4@00018000?o00006`03oaD0 0_l601go2P04o`D000?o00000P02o`8000?o00004@05oa800_l20003o`00008000?o0?l04P03o`05 0?oo0?l040001Ol0003o00D000?o00001005o`030?oo00Go00@0o`001_l>00004P000ol0000N00?o 3`03o`801_lH00oo20000ol000030003o`3o008000?o00003@03o`040?l000Co3P000ol000020003 o`00008000Co003o40001?l00?l200;o00@0oooo2`02o`<000?o00001@000ol0000700?o1P001_l0 003o00;o0P04o`X0000B0003o`0002401?l900Oo7@001?l0ool20003o`0000h000?o00001@02o`<0 0_l<0004o`0000;o0`001?l00?l<0003o`00008000?o000010000ol00002o``00_l20007o`000?l0 o`0300;o1P03o`@00_l80003o`0000d00_l500;o00<0ool03P0001800_lV01Co5`02o`040?l0o`<0 00?o000030000ol000060003o`3o00@00_l800;o0`02o`D000?o00000ol600?o10000ol000060005 o`000?l02P001Ol0003o00D00_l500Ko1P02o`X00_l<0003o`0000D000?o0?l00P000ol0000;0000 4P000ol0000Z00Go00<0ool01?l300;o50001?l00002o`D00_l:00;o2P000ol0o`0500So10000ol0 o`050003o`0000801ol60003o`0000L000?o00000P02o`H00_l30003o`0000@000?o0?l03`000ol0 o`0<0003o`0000P000?o00001`000ol0o`020003o`0000X0000B0003o`0002L01ol400?o00<0o`00 0`03o`l00_l30003o`3o00L00ol400?o30001?l00?l?0003o`3o00L000?o00003@000ol000090003 o`0000<01_l40003o`0000H000Co0?oo30000ol0o`0>00;o1`000ol0000900;o0`000ol000090000 4P000ol0000U00;o0`02o`L00ol00`3o00050003o`0000X00_l40003o`3o00X01Ol@0004o`00o`d0 00Co003o20000ol0000;0003o`0000/000?o00002`000ol000070005o`000?l02P001?l00?l@00Oo 30000ol0o`030003o`0000P0000B0003o`0002@000Co00001?l900;o00<0o`001@05o`801Ol50004 o`00ob0000?o00000_l:0004o`00o`X00_l900;o3`000ol000090003o`0000T000Co00000_l600;o 0P000ol0000S0003o`3o00d0000B00;o8`02o`<000?o0?l00_l900;o0P000ol0000700Co2@001?l0 oolP0003o`00008000?o00001P001Ol0003o00`000?o00001@000ol0000@00;o2@000ol0000:0004 o`0000Oo10000ol0000S0004o`00o``0000B0003o`00024000?o00000P000ol00002o`/00_l20003 o`00018000Co003o8P001?l00002o`D00_l30003o`0000`00ol200;o5@000ol000050003o`0000`0 00?o00002P000ol0000U0003o`3o00`0000B0003o`00020000?o00000P000ol00003o`/00_l300;o 4@000ol00002ob<000?o00000P05o`@000?o00003`03oaL02?l?0003o`0000T000?o00009P001?l0 0?l;00004P000ol0000O0003o`0000<000Go0?l0o`0=00;o10000ol0000=0003o`0000;o9@000ol0 000:0003o`0002`000?o00004P000ol000070003o`0002P000Co003o2P00018000?o00007@02o`D0 00?o00000ol=00;o1@02o``000Co00000_lU0003o`0000T000?o0000@`02o`L000?o0000:0000ol0 000;00004P000ol0000L0003o`0000D000Co0?l00_l>00;o1P000ol0000600?o0`000ol0o`0V0003 o`0000H00_m80003o`0000800_l/0003o`0000X0000B00;o70000ol000050004o`3o00;o3`000ol0 o`060003o`0000@000?o00001002obP00_l50003o`0004T01?l_0003o`0000T0000B0003o`0001T0 0_l80004o`3o00;o40000ol0o`0600Ko1P000ol0o`0Y00;o0P03ogh000?o00002@00018000?o0000 60000ol000070004o`3o00;o4@001?l00?l@0003o`3o02`00on10003o`0000P0000B0003o`0001H0 0_l90003o`0000;o00<0o`004@001?l00?l>0004o`00ok/0000B0003o`0001D000?o00002@000ol0 0003oa@000Co003o30001Ol0003o0;/0000B00;o5@000ol000090005o`00o`000_lD0005o`000?l0 2`001?l00?nl00004P000ol0000B00;o30000ol00002o`030?l001@000Go0000o`090005o`000?l0 _000018000?o00004@000ol0000;0004o`0000?o5P000ol000020003o`0000@00_l30003o`000;/0 000B0003o`0000l00_l=0006o`000?l00_lG0003o`0000801_l50003o`000;/0000:00?o1@000ol0 000<00?o3`001_l0003o00;o5`000ol0000=0003o`000;/0000;0003o`0000@000?o00002@03oa40 00?o00000P001?l0oolG0003o`0000`000?o0000_00000/000?o0000100P0000B0003o`0001D000?o000010001ol0003o0?l0j000018000?o00005@000ol0 00040006o`00o`3oj@00018000?o000050000ol000040007o`000?l0o`3Y00004P000ol0000D0003 o`0000@000Oo0000o`3o0>T0000B00;o50000ol000050004o`00o`8000?o0000i`00018000?o0000 4`000ol000040005o`000?l00P000ol0003W00004P000ol0000B0003o`0000D000Oo0000o`3o0>X0 000B0003o`00014000?o00001P001?l00?l20003o`000>P0000B0003o`00010000?o00001P001Ol0 003o008000?o0000j00001800_lA0003o`0000H000Go0000o`020003o`000>P0000B0003o`0000h0 0_l90007o`000?l0o`3[00004P000ol0000<00;o2P001Ol0003o008000?o0000j@00018000?o0000 2`000ol0000:0005o`000?l00P000ol0003Y00002P04o`@000?o00002P000ol0000;0005o`000?l0 0P000ol0003Y00002`000ol000040003o`0000H01?l=0003o`00008000?o0?l0k00000`000?o0000 0`0:oa0000Go0000o`020003o`000>X0000=0003o`00008000?o00005`001Ol0003o008000?o0000 jP0000X000Co003o10000ol0000F0003o`00008000Co003ok00000/00_l50003o`0001H000?o0000 0P001?l00?o/00004P000ol0000F0005o`000?l00P000ol0003[00004P000ol0000E0003o`000080 00Co003ok@0001800_lF0003o`00008000Co003ok@00018000?o00005@000ol000020004o`00ond0 000B0003o`0001@000?o00000P001?l00?o^00004P000ol0000D0003o`00008000Co003okP000180 00?o000050000ol000020004o`00onh0000B00;o50000ol000030004o`00onh0000B0003o`0001<0 00?o00000P001Ol0003o0>h0000B0003o`00018000?o00000`001?l00?o_00004P000ol0000B0003 o`0000<000Co003ok`00018000?o00004@000ol000040004o`00onl0000B0003o`00014000?o0000 0`001Ol0003o0>l0000B00;o4@000ol000040005o`000?l0k`00018000?o000040000ol000040004 o`00oo00000B0003o`0000l000?o00001@001?l00?o`00004P000ol0000?0003o`0000@000Go0000 o`3`00004P000ol0000>0003o`0000D000Go0000o`3`00004P02o`h000?o00001P001Ol0003o0?00 000B0003o`0000`000?o00001`001?l00?oa00004P000ol0000;0003o`0000L000Go0000o`3a0000 4P000ol0000:0003o`0000P000Go0000o`3a00002P03o`D000?o00002@000ol000090005o`000?l0 l@0000d000?o00000P000ol0000600?o30001Ol0003o0?40000=0003o`0000802Ol>0005o`000?l0 lP0000/00_l50003o`0001@000Go0000o`3b00003@000ol000020003o`0001@000Go0000o`3b0000 2P03o`D000?o000050001Ol0003o0?80000B0003o`0001<000?o00000P000ol0003`00004P000ol0 000C0005o`000?l0l`0001800_lD0005o`000?l0l`00018000?o00004P000ol000020003o`000?40 000B0003o`00018000?o00000P000ol0003a00004P000ol0000B0003o`00008000?o0000l@000180 00?o00004@000ol000020003o`000?80000B00;o4P000ol000020003o`000?80000B0003o`000140 00?o00000P000ol0003b00004P000ol0000@0003o`0000<000?o0000lP00018000?o000040000ol0 00030003o`000?80000B0003o`00010000?o00000P000ol0003c00004P000ol0000?0003o`0000<0 00?o0000l`0001800_l@0003o`0000<000?o0000l`00018000?o00003`000ol000030003o`000?<0 000B0003o`0000h000?o000010000ol0003c00004P000ol0000=0003o`0000@000?o0000m0000180 00?o00003@000ol000040003o`000?@0000B00;o3@000ol000050003o`000?@0000B0003o`0000/0 00?o00001P000ol0003d00004P000ol0000:0003o`0000L000?o0000m000018000?o00002@000ol0 00070003o`000?D0000;00?o10000ol0000700;o2P000ol0003e000030000ol000030003o`0000D0 0_l<0003o`000?D0000:00Co1008o`h000?o0000m@0000X000?o0?l01@000ol0000C0003o`000?D0 000:0003o`3o00D000?o00004P000ol0003f00002`02o`D000?o00004P000ol0003f00004P000ol0 000B0003o`000?H0000B0003o`00014000?o0000m`0001800_lB0003o`000?L0000B0003o`000140 00?o0000m`00018000?o000040000ol0003h00004P000ol0000@0003o`000?P0000B0003o`000100 00?o0000n00001800_lA0003o`000?P0000B0003o`0000l000?o0000n@00018000?o00003`000ol0 003i00004P000ol0000?0003o`000?T0000B0003o`0000l000?o0000n@00018000?o00003P000ol0 003j00004P02o`l000?o0000nP00018000?o00003@000ol0003k00004P000ol0000=0003o`000?/0 000B0003o`0000`000?o0000o000018000?o00002`000ol0003m00004P02o`/000?o0000oP000180 00?o00002P000ol0003n00004P000ol000090003o`000?l0000B0003o`0000T000?o0000o`0000X0 0ol50003o`0000L00_oo00<0000=0003o`00008000?o00001003ool01@0000d000?o00000P08ool0 1`0000X00ooo01@0000:0003o`000?l0500000X01?oo01<0003o02400000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.07632, -1.1608, 0.0579624, 0.0357276}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "The dominant terms for large values of ", StyleBox["t", "Input"], " for all these solutions are" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(-\(75\/625\)\)\ y0\ \ Cos[2\ t] + \(800\/5000\) y0\ Sin[2\ t] + t\ \((3\/100\ Cos[2\ t] - 1\/25\ Sin[2\ t])\) // Simplify\)], "Input"], Cell[BoxData[ \(1\/100\ \((t - 4\ y0)\)\ \((3\ Cos[2\ t] - 4\ Sin[2\ t])\)\)], "Output"] }, Open ]], Cell[TextData[{ "This expression vanishes (for any ", StyleBox["y0", "Input"], ") when ", StyleBox["3 Cos[2 t]-4 Sin[2 t] == 0", "Input"], ", i.e., when\n", StyleBox["t == (ArcTan[3/4] + k*Pi)/2", "Input"], ". The interval between successive zeros of the solutions is then \ approximately ", StyleBox["Pi/2", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["Problem 6", "Subsubsection"], Cell[TextData[{ "Find approximate values of the solution of the initial value problem\n", StyleBox["\ty'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1\n", "Input"], "at ", StyleBox["t==1", "Input"], " using the Euler method with ", StyleBox["h = 0.1", "Input"], ", ", StyleBox["h = 0.05", "Input"], ", and ", StyleBox["h = 0.025", "Input"], ". Compare these results with the output of ", StyleBox["NDSolve[]", "Input"], " by superimposing plots of the data." }], "Text"], Cell["Solution", "Subsubsection"], Cell[TextData[{ "f[t_,y_] = ", StyleBox["(t^2 - y^2) Sin[y];", "Input"] }], "Input"], Cell["EMStep[{t_,y_}] := N[{t+h,y+h*f[t,y]}];", "Input"] }, Open ]], Cell["h = 0.1;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "To reach the point ", StyleBox["t==1.0", "Input"], " with this step size we need ", StyleBox["10", "Input"], " steps." }], "Text"], Cell[CellGroupData[{ Cell["soln1=NestList[EMStep,{0,1},10]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.1`, 0.915852901519210327`}, {0.2`, 0.850123300757828559`}, { 0.300000000000000044`, 0.798827109022698067`}, {0.4`, 0.759551863869434296`}, {0.5`, 0.7308429630007927`}, {0.6`, 0.711877259152616304`}, {0.7`, 0.702289467138418821`}, { 0.799999999999999911`, 0.70208207968867109`}, {0.9`, 0.711580683468661234`}, {0.999999999999999822`, 0.731410182020548393`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ p1 = ListPlot[soln1,PlotJoined->True, PlotStyle->RGBColor[0,1,0]]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.37241 1.97572 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .0106 -18 -4.5 ] [.01131 .0106 0 4.5 ] [.01131 .10939 -24 -4.5 ] [.01131 .10939 0 4.5 ] [.01131 .20817 -18 -4.5 ] [.01131 .20817 0 4.5 ] [.01131 .30696 -24 -4.5 ] [.01131 .30696 0 4.5 ] [.01131 .40575 -18 -4.5 ] [.01131 .40575 0 4.5 ] [.01131 .50453 -24 -4.5 ] [.01131 .50453 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .0106 m .03006 .0106 L s [(0.7)] .01131 .0106 1 0 Mshowa .02381 .10939 m .03006 .10939 L s [(0.75)] .01131 .10939 1 0 Mshowa .02381 .20817 m .03006 .20817 L s [(0.8)] .01131 .20817 1 0 Mshowa .02381 .30696 m .03006 .30696 L s [(0.85)] .01131 .30696 1 0 Mshowa .02381 .40575 m .03006 .40575 L s [(0.9)] .01131 .40575 1 0 Mshowa .02381 .50453 m .03006 .50453 L s [(0.95)] .01131 .50453 1 0 Mshowa .125 Mabswid .02381 .03036 m .02756 .03036 L s .02381 .05012 m .02756 .05012 L s .02381 .06987 m .02756 .06987 L s .02381 .08963 m .02756 .08963 L s .02381 .12914 m .02756 .12914 L s .02381 .1489 m .02756 .1489 L s .02381 .16866 m .02756 .16866 L s .02381 .18842 m .02756 .18842 L s .02381 .22793 m .02756 .22793 L s .02381 .24769 m .02756 .24769 L s .02381 .26745 m .02756 .26745 L s .02381 .2872 m .02756 .2872 L s .02381 .32672 m .02756 .32672 L s .02381 .34647 m .02756 .34647 L s .02381 .36623 m .02756 .36623 L s .02381 .38599 m .02756 .38599 L s .02381 .4255 m .02756 .4255 L s .02381 .44526 m .02756 .44526 L s .02381 .46502 m .02756 .46502 L s .02381 .48478 m .02756 .48478 L s .02381 .52429 m .02756 .52429 L s .02381 .54405 m .02756 .54405 L s .02381 .5638 m .02756 .5638 L s .02381 .58356 m .02756 .58356 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 1 0 r .5 Mabswid .02381 .60332 m .11905 .43707 L .21429 .3072 L .30952 .20586 L .40476 .12826 L .5 .07154 L .59524 .03407 L .69048 .01512 L .78571 .01472 L .88095 .03348 L .97619 .07266 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z0200;L/`0001/000?o0000C0000m`0 002c00006`000ol0001;0003g0000;@0000800;o1002o`@00_l50003o`0004X000?L0000]@0000L0 00Co003o0`02o`<000Co003o10000ol000190003g0000;H000070004o`00o`P000Co003o1002odX0 00?L0000]P0000L000Co003o2@02o`D000?o0000B0000m`0002g00001`001?l00?l80004o`00o`@0 00?o0000A`000m`0002h00002002o`X00_l50003o`0004H000?L0000^@0001/000?o0000A@000m`0 002j00006`02odD000?L0000^`0001/000?o0000@`000m`0002l00006`000ol000120003g0000;d0 000K0003o`00044000?L0000_P0001/000?o0000@0000m`0002o00006`000ol0000o0003g0000<00 000K00;o?`000m`0003100006`000ol0000m0003g0000<80000K0003o`0003`000?L0000``0001/0 00?o0000>`000m`0003400006`000ol0000j0003g0000P000m`0003600006`000ol0 000h0003g000040000800;o1002o`<0 0ol50003o`0001d000?L0000hP0000L000Co003o0`02o`H000?o00000P000ol0000M0003g0000>80 00070004o`00o`T00ol400;o7@000m`0003S00001`001?l00?l80004o`00o`@000?o00006`000m`0 003T00001`001?l00?l80004o`00o`@000?o00006`000m`0003T00002002o`X00_l50003o`0001X0 00?L0000i@0001/000?o00006@000m`0003V00006`02oaT000?L0000i`0001/000?o000060000m`0 003W00006`000ol0000G0003g0000>P0000K0003o`0001H000?L0000j@0001/000?o00005P000m`0 003Y00006`02oaH000?L0000jP0001/000?o00005@000m`0003Z00006`000ol0000D0003g0000>/0 000K0003o`0001<000?L0000k00001/000?o00004`000m`0003/00006`02oa<000?L0000k@0001/0 00?o00004P000m`0003]00006`000ol0000A0003g0000>h0000K0003o`00014000?L0000kP0001/0 00?o000040000m`0003_00006`000ol0000?0003g0000?00000K00;o40000m`0003`00006`000ol0 000>0003g0000?40000K0003o`0000h000?L0000l@0000800_l400;o0`03o`<00ol50003o`0000d0 00?L0000lP0000050?l00?l00`02o`H000?o00000`000ol000020003o`0000d000?L0000lP000005 0?l00?l02@03o`D000?o00000P02o`d000?L0000l`0000050?l00?l020001?l00?l200?o1@000ol0 000;0003g0000?@000001@3o003o00P000Co003o0P000ol000050003o`0000/000?L0000m0000080 0_l:00;o0`04o`@000?o00002P000m`0003e00006`000ol0000:0003g0000?D0000K00;o2P000m`0 003f00006`000ol000090003g0000?H0000K0003o`0000P000?L0000m`0001/000?o000020000m`0 003g00006`000ol000070003g0000?P0000K00;o1`000m`0003i00006`000ol000060003g0000?T0 000K0003o`0000D000?L0000nP0001/000?o00001@000m`0003j00006`000ol000040003g0000?/0 000K0003o`0000@000?L0000n`0001/00_l40003g00002800_l400;o0`04ob@00_l400;o1003ob@0 0_l400;o1002ob@00_l400;o1002obX00ol700006`000ol000020003g000028000Co003o0`02o`@0 00?o00008`001?l00?l300;o1@000ol0000R0004o`00o`<00_l30004o`00ob8000Co003o0`02o`<0 00Co003o:P000ol0000600006`000ol000020003g000028000Co003o2P000ol0000R0004o`00o`P0 1?lS0004o`00o`P000Co003o8P001?l00?l80004o`00obX000?o00001P0001/000Go0000g00U0004 o`00o`/000?o00008@001?l00?l80003o`3o02@000Co003o2003ob<000Co003o2@02ob/000?o0000 1P0001/000Go0000g00U0004o`00o`P000Co003o8`001?l00?l80003o`3o02@000Co003o20000ol0 000S0004o`00o`P000Co003o:P000ol0000600006`02o`030=`002H00_l:00;o9@02o`X00_lU00;o 2P03ob<00_l:00;o:P02o`P0000K0004o`00g?l00P0001/000?o0=`0o`0300006`000ooL003o00<0 000K0003om`00?l00`0001@01ol00mcoo`3oo`?o000K0003o`0000T000?o00002P000ol0000:0003 o`0000T000?o00002P000ol0000:0003o`0000T000?o00002P000ol0000:0003o`0000T000?o0000 2P000ol0000:0003o`0000T000?o00002P000ol0000:0003o`0000T000?o00002P000ol0000:0003 o`0000T000?o00002P000ol0000500006`000ol0000_0003o`00030000?o0000<0000ol0000_0003 o`00030000?o00001@0001/000?o0000o`0300006`000ol0003o00<0003o0240003o0240003o0240 003o0240003o02400000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.681111, 0.00394728, 0.00190276}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["h = 0.05;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["Twenty steps are neede for this case.", "Text"], Cell[CellGroupData[{ Cell["soln2=NestList[EMStep,{0,1},20]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.05`, 0.957926450759605074`}, {0.1`, 0.920497889761777088`}, {0.150000000000000017`, 0.887176753008737861`}, {0.2`, 0.85753802741485483`}, {0.25`, 0.831244850596224615`}, {0.3`, 0.808030156451488146`}, {0.35`, 0.787682899828316784`}, {0.399999999999999964`, 0.770037731696357496`}, {0.45`, 0.754967275509267654`}, { 0.499999999999999911`, 0.742376370613666303`}, { 0.549999999999999911`, 0.732197808666120497`}, {0.6`, 0.724389205607113062`}, {0.65`, 0.718930735007772714`}, { 0.700000000000000088`, 0.71582350644893804`}, {0.750000000000000088`, 0.715088410718678257`}, {0.800000000000000177`, 0.716765275939706825`}, {0.850000000000000177`, 0.720912187832911133`}, {0.900000000000000177`, 0.727604824839567498`}, {0.950000000000000355`, 0.736935645926333204`}, {1.00000000000000022`, 0.749012746749339175`}} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ p2 = ListPlot[soln2,PlotJoined->True, PlotStyle->RGBColor[0,0,1]]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.4626 2.06592 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .08684 -24 -4.5 ] [.01131 .08684 0 4.5 ] [.01131 .19014 -18 -4.5 ] [.01131 .19014 0 4.5 ] [.01131 .29343 -24 -4.5 ] [.01131 .29343 0 4.5 ] [.01131 .39673 -18 -4.5 ] [.01131 .39673 0 4.5 ] [.01131 .50002 -24 -4.5 ] [.01131 .50002 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .08684 m .03006 .08684 L s [(0.75)] .01131 .08684 1 0 Mshowa .02381 .19014 m .03006 .19014 L s [(0.8)] .01131 .19014 1 0 Mshowa .02381 .29343 m .03006 .29343 L s [(0.85)] .01131 .29343 1 0 Mshowa .02381 .39673 m .03006 .39673 L s [(0.9)] .01131 .39673 1 0 Mshowa .02381 .50002 m .03006 .50002 L s [(0.95)] .01131 .50002 1 0 Mshowa .125 Mabswid .02381 .1075 m .02756 .1075 L s .02381 .12816 m .02756 .12816 L s .02381 .14882 m .02756 .14882 L s .02381 .16948 m .02756 .16948 L s .02381 .21079 m .02756 .21079 L s .02381 .23145 m .02756 .23145 L s .02381 .25211 m .02756 .25211 L s .02381 .27277 m .02756 .27277 L s .02381 .31409 m .02756 .31409 L s .02381 .33475 m .02756 .33475 L s .02381 .35541 m .02756 .35541 L s .02381 .37607 m .02756 .37607 L s .02381 .41739 m .02756 .41739 L s .02381 .43805 m .02756 .43805 L s .02381 .4587 m .02756 .4587 L s .02381 .47936 m .02756 .47936 L s .02381 .52068 m .02756 .52068 L s .02381 .54134 m .02756 .54134 L s .02381 .562 m .02756 .562 L s .02381 .58266 m .02756 .58266 L s .02381 .06618 m .02756 .06618 L s .02381 .04552 m .02756 .04552 L s .02381 .02486 m .02756 .02486 L s .02381 .0042 m .02756 .0042 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .5 Mabswid .02381 .60332 m .07143 .5164 L .11905 .43907 L .16667 .37024 L .21429 .309 L .2619 .25468 L .30952 .20673 L .35714 .16469 L .40476 .12824 L .45238 .0971 L .5 .07109 L .54762 .05006 L .59524 .03393 L .64286 .02265 L .69048 .01623 L .7381 .01472 L .78571 .01818 L .83333 .02675 L .88095 .04057 L .92857 .05985 L .97619 .0848 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z02P000m80003500006`02ocX000?B0000aP0001/000?o0000 >0000m80003700006`000ol0000g0003dP0000000070004o`00o`T00ol400;o 80000m80003P00001`001?l00?l80004o`00o`@000?o00007P000m80003Q00001`001?l00?l80004 o`00o`@000?o00007@000m80003R00002002o`X00_l50003o`0001`000?B0000h`0001/000?o0000 70000m80003S00006`000ol0000K0003dP000>@0000K00;o6`000m80003U00006`000ol0000J0003 dP000>D0000K0003o`0001T000?B0000iP0001/000?o000060000m80003W00006`000ol0000H0003 dP000>L0000K00;o60000m80003X00006`000ol0000F0003dP000>T0000K0003o`0001H000?B0000 j@0001/000?o00005@000m80003Z00006`000ol0000D0003dP000>/0000K0003o`0001@000?B0000 j`0001/00_lD0003dP000>`0000K0003o`00018000?B0000k@0001/000?o00004P000m80003]0000 6`000ol0000A0003dP000>h0000K0003o`00010000?B0000k`0001/00_lA0003dP000>l0000K0003 o`0000l000?B0000l00001/000?o00003`000m80003`00006`000ol0000>0003dP000?40000200;o 1002o`<00ol300?o1@000ol0000=0003dP000?8000001@3o003o00<00_l60003o`0000<000?o0000 0P000ol0000=0003dP000?8000001@3o003o00T00ol50003o`0000800_l=0003dP000?<000001@3o 003o00P000Co003o0P03o`D000?o00002`000m80003d000000D0o`00o`080004o`00o`8000?o0000 1@000ol0000;0003dP000?@0000200;o2P02o`<01?l40003o`0000X000?B0000m@0001/000?o0000 2@000m80003f00006`02o`X000?B0000mP0001/000?o000020000m80003g00006`000ol000080003 dP000?L0000K0003o`0000L000?B0000n00001/000?o00001`000m80003h00006`000ol000060003 dP000?T0000K00;o1`000m80003i00006`000ol000050003dP000?X0000K0003o`0000D000?B0000 nP0001/000?o000010000m80003k00006`000ol000040003dP000?/0000K00;o10000m80003l0000 6`000ol000020003dP0002<00_l400;o0`04ob@00_l400;o1003ob@00_l400;o1002ob@00_l400;o 1002obX00ol700006`000ol000020003dP00028000Co003o0`02o`@000?o00008`001?l00?l300;o 1@000ol0000R0004o`00o`<00_l30004o`00ob8000Co003o0`02o`<000Co003o:P000ol000060000 6`001Ol0003B02D000Co003o2P000ol0000R0004o`00o`P01?lS0004o`00o`P000Co003o8P001?l0 0?l80004o`00obX000?o00001P0001/000Go0000dP0U0004o`00o`/000?o00008@001?l00?l80003 o`3o02@000Co003o2003ob<000Co003o2@02ob/000?o00001P0001/000Co003B9P001?l00?l80004 o`00ob<000Co003o20000ol0o`0T0004o`00o`P000?o00008`001?l00?l80004o`00obX000?o0000 1P0001/00_l00`3B000V00;o2P02obD00_l:00;o9@02o`X00olS00;o2P02obX00_l800006`000ol0 dP3o00<0000K0003o`3B0?l00`0001/000?odP00o`0300006`000ooB003o00<0000D00Oo00?Bool0 ool3o`006`000ol000090003o`0000X000?o00002P000ol000090003o`0000X000?o00002P000ol0 00090003o`0000X000?o00002P000ol000090003o`0000X000?o00002P000ol000090003o`0000X0 00?o00002P000ol000090003o`0000X000?o00002P000ol000090003o`0000X000?o00001@0001/0 00?o0000;`000ol0000`0003o`00030000?o0000;`000ol0000`0003o`0000D0000K0003o`000?l0 0`0001/000?o0000o`030000o`0Q0000o`0Q0000o`0Q0000o`0Q0000o`0Q0000o`0Q0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.696558, 0.00394728, 0.00181968}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["h = 0.025;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["soln3=NestList[EMStep,{0,1},40]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.025`, 0.978963225379802537`}, {0.05`, 0.959091966886821367`}, {0.0750000000000000088`, 0.940316631854297568`}, {0.1`, 0.922575115573851256`}, {0.125`, 0.905811960223883971`}, {0.15`, 0.889977617249906494`}, {0.175`, 0.875027801848675323`}, {0.199999999999999982`, 0.860922928918173724`}, {0.225`, 0.847627620744331444`}, { 0.249999999999999955`, 0.835110277672112921`}, { 0.274999999999999955`, 0.823342703975641043`}, {0.3`, 0.812299782055863417`}, {0.325`, 0.801959188932828404`}, { 0.350000000000000044`, 0.792301149754081102`}, { 0.375000000000000044`, 0.783308223710053486`}, { 0.400000000000000088`, 0.774965118335141589`}, { 0.425000000000000088`, 0.767258528685285323`}, { 0.450000000000000088`, 0.760176998326143493`}, { 0.475000000000000177`, 0.753710799447383728`}, { 0.500000000000000088`, 0.74785182974491704`}, {0.525000000000000177`, 0.742593523990254311`}, {0.550000000000000177`, 0.737930778440054258`}, {0.575000000000000177`, 0.733859886434158603`}, {0.600000000000000177`, 0.73037848369108822`}, {0.625000000000000177`, 0.727485501939553103`}, {0.650000000000000266`, 0.725181129625892673`}, {0.675000000000000266`, 0.723466778512863051`}, {0.700000000000000266`, 0.72234505503671631`}, {0.725000000000000355`, 0.721819735318613986`}, {0.750000000000000355`, 0.721895742734292689`}, {0.775000000000000355`, 0.722579126933580528`}, {0.800000000000000355`, 0.723877043169727052`}, {0.825000000000000355`, 0.725797730748434144`}, {0.850000000000000355`, 0.728350489338935958`}, {0.875000000000000355`, 0.731545651805680918`}, {0.900000000000000532`, 0.735394552120746869`}, {0.925000000000000532`, 0.739909486806296623`}, {0.950000000000000532`, 0.745103668236182414`}, {0.975000000000000532`, 0.75099116800038832`}, {1.00000000000000044`, 0.757586848410821378`}} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ p3 = ListPlot[soln3,PlotJoined->True, PlotStyle->RGBColor[1,0,0]]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.51259 2.11591 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .07434 -24 -4.5 ] [.01131 .07434 0 4.5 ] [.01131 .18014 -18 -4.5 ] [.01131 .18014 0 4.5 ] [.01131 .28593 -24 -4.5 ] [.01131 .28593 0 4.5 ] [.01131 .39173 -18 -4.5 ] [.01131 .39173 0 4.5 ] [.01131 .49752 -24 -4.5 ] [.01131 .49752 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .07434 m .03006 .07434 L s [(0.75)] .01131 .07434 1 0 Mshowa .02381 .18014 m .03006 .18014 L s [(0.8)] .01131 .18014 1 0 Mshowa .02381 .28593 m .03006 .28593 L s [(0.85)] .01131 .28593 1 0 Mshowa .02381 .39173 m .03006 .39173 L s [(0.9)] .01131 .39173 1 0 Mshowa .02381 .49752 m .03006 .49752 L s [(0.95)] .01131 .49752 1 0 Mshowa .125 Mabswid .02381 .0955 m .02756 .0955 L s .02381 .11666 m .02756 .11666 L s .02381 .13782 m .02756 .13782 L s .02381 .15898 m .02756 .15898 L s .02381 .2013 m .02756 .2013 L s .02381 .22246 m .02756 .22246 L s .02381 .24361 m .02756 .24361 L s .02381 .26477 m .02756 .26477 L s .02381 .30709 m .02756 .30709 L s .02381 .32825 m .02756 .32825 L s .02381 .34941 m .02756 .34941 L s .02381 .37057 m .02756 .37057 L s .02381 .41289 m .02756 .41289 L s .02381 .43405 m .02756 .43405 L s .02381 .45521 m .02756 .45521 L s .02381 .47636 m .02756 .47636 L s .02381 .51868 m .02756 .51868 L s .02381 .53984 m .02756 .53984 L s .02381 .561 m .02756 .561 L s .02381 .58216 m .02756 .58216 L s .02381 .05318 m .02756 .05318 L s .02381 .03202 m .02756 .03202 L s .02381 .01086 m .02756 .01086 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .5 Mabswid .02381 .60332 m .04762 .55881 L .07143 .51676 L .09524 .47703 L .11905 .43949 L .14286 .40403 L .16667 .37052 L .19048 .33889 L .21429 .30904 L .2381 .28091 L .2619 .25443 L .28571 .22953 L .30952 .20616 L .33333 .18428 L .35714 .16385 L .38095 .14482 L .40476 .12717 L .42857 .11086 L .45238 .09588 L .47619 .08219 L .5 .0698 L .52381 .05867 L .54762 .0488 L .57143 .04019 L .59524 .03282 L .61905 .0267 L .64286 .02183 L .66667 .0182 L .69048 .01583 L .71429 .01472 L .7381 .01488 L .7619 .01632 L .78571 .01907 L .80952 .02313 L .83333 .02853 L .85714 .03529 L .88095 .04344 L .90476 .05299 L .92857 .06398 L .95238 .07644 L .97619 .0904 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z021008S 2@0000800_l900Co0P04o`@000?o0000LP028hD000I00006`000ol0001V00038`0009T0000K00;oIP000b<0002J0000 6`000ol0001T00038`0009/0000K0003o`0006<0000003 8`000;40000800;o2P02o`D000?o0000C@000b<0002b00006`000ol0001<00038`000;<0000K00;o C0000b<0002d00006`000ol0001:00038`000;D0000K0003o`0004P00R>h00006`000ol000170003 8`000;P0000K0003o`0004H0000000b<0003700000P02o`@00_l400;o0`03o`D000?o0000 =`000b<00038000000D0o`00o`0300;o0`001?l00?l50003o`00008000?o0000=P000b<000390000 00D0o`00o`080004o`00o`D000?o00000P02ocH000<0000K0003 o`0001`000H0000K0003o`0001T000T0000K0003o`0001H000/0000K00;o5@000b<0003[00006`000ol0000C00038`000>`0000K0003o`0001<000h0000K0003o`00014000l0000K0003o`00010000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.703727, 0.00394728, 0.00177669}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "soln = NDSolve[{y'[t]==", StyleBox["(t^2-y[t]^2) Sin[y[t]]", "Input"], ",y[0] == 1}, y[t],\n {t,0,1}]" }], "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{\(y[t]\), "\[Rule]", RowBox[{ TagBox[\(InterpolatingFunction[{{0.`, 1.`}}, "<>"]\), False, Editable->False], "[", "t", "]"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell["y[t_] = y[t]/.First[soln];", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["y[1.0]", "Input"], Cell[BoxData[ \(0.766013433256353604`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["p = Plot[y[t],{t,0,1}]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -2.87391 3.9661 [ [.21429 .28647 -9 -9 ] [.21429 .28647 9 0 ] [.40476 .28647 -9 -9 ] [.40476 .28647 9 0 ] [.59524 .28647 -9 -9 ] [.59524 .28647 9 0 ] [.78571 .28647 -9 -9 ] [.78571 .28647 9 0 ] [.97619 .28647 -3 -9 ] [.97619 .28647 3 0 ] [.01131 .061 -24 -4.5 ] [.01131 .061 0 4.5 ] [.01131 .14033 -24 -4.5 ] [.01131 .14033 0 4.5 ] [.01131 .21965 -24 -4.5 ] [.01131 .21965 0 4.5 ] [.01131 .37829 -24 -4.5 ] [.01131 .37829 0 4.5 ] [.01131 .45761 -24 -4.5 ] [.01131 .45761 0 4.5 ] [.01131 .53694 -24 -4.5 ] [.01131 .53694 0 4.5 ] [.01131 .61626 -24 -4.5 ] [.01131 .61626 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .29897 m .21429 .30522 L s [(0.2)] .21429 .28647 0 1 Mshowa .40476 .29897 m .40476 .30522 L s [(0.4)] .40476 .28647 0 1 Mshowa .59524 .29897 m .59524 .30522 L s [(0.6)] .59524 .28647 0 1 Mshowa .78571 .29897 m .78571 .30522 L s [(0.8)] .78571 .28647 0 1 Mshowa .97619 .29897 m .97619 .30522 L s [(1)] .97619 .28647 0 1 Mshowa .125 Mabswid .07143 .29897 m .07143 .30272 L s .11905 .29897 m .11905 .30272 L s .16667 .29897 m .16667 .30272 L s .2619 .29897 m .2619 .30272 L s .30952 .29897 m .30952 .30272 L s .35714 .29897 m .35714 .30272 L s .45238 .29897 m .45238 .30272 L s .5 .29897 m .5 .30272 L s .54762 .29897 m .54762 .30272 L s .64286 .29897 m .64286 .30272 L s .69048 .29897 m .69048 .30272 L s .7381 .29897 m .7381 .30272 L s .83333 .29897 m .83333 .30272 L s .88095 .29897 m .88095 .30272 L s .92857 .29897 m .92857 .30272 L s .25 Mabswid 0 .29897 m 1 .29897 L s .02381 .061 m .03006 .061 L s [(0.74)] .01131 .061 1 0 Mshowa .02381 .14033 m .03006 .14033 L s [(0.76)] .01131 .14033 1 0 Mshowa .02381 .21965 m .03006 .21965 L s [(0.78)] .01131 .21965 1 0 Mshowa .02381 .37829 m .03006 .37829 L s [(0.82)] .01131 .37829 1 0 Mshowa .02381 .45761 m .03006 .45761 L s [(0.84)] .01131 .45761 1 0 Mshowa .02381 .53694 m .03006 .53694 L s [(0.86)] .01131 .53694 1 0 Mshowa .02381 .61626 m .03006 .61626 L s [(0.88)] .01131 .61626 1 0 Mshowa .125 Mabswid .02381 .08084 m .02756 .08084 L s .02381 .10067 m .02756 .10067 L s .02381 .1205 m .02756 .1205 L s .02381 .16016 m .02756 .16016 L s .02381 .17999 m .02756 .17999 L s .02381 .19982 m .02756 .19982 L s .02381 .23948 m .02756 .23948 L s .02381 .25931 m .02756 .25931 L s .02381 .27914 m .02756 .27914 L s .02381 .3188 m .02756 .3188 L s .02381 .33863 m .02756 .33863 L s .02381 .35846 m .02756 .35846 L s .02381 .39812 m .02756 .39812 L s .02381 .41795 m .02756 .41795 L s .02381 .43778 m .02756 .43778 L s .02381 .47745 m .02756 .47745 L s .02381 .49728 m .02756 .49728 L s .02381 .51711 m .02756 .51711 L s .02381 .55677 m .02756 .55677 L s .02381 .5766 m .02756 .5766 L s .02381 .59643 m .02756 .59643 L s .02381 .04117 m .02756 .04117 L s .02381 .02134 m .02756 .02134 L s .02381 .00151 m .02756 .00151 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .18652 .61803 m .22272 .53463 L .26171 .45319 L .30316 .37516 L .34309 .3079 L .3815 .25019 L .42237 .19608 L .46172 .15085 L .49955 .11358 L .53984 .08048 L .57861 .05499 L .59832 .04441 L .61984 .03469 L .64066 .0271 L .65954 .02177 L .66899 .01966 L .67922 .0178 L .68484 .01696 L .69002 .0163 L .69493 .01579 L .70019 .01534 L .7028 .01517 L .70528 .01503 L .70769 .01492 L .70993 .01483 L .71204 .01478 L .7131 .01475 L .71428 .01474 L .71545 .01472 L .71672 .01472 L .71792 .01472 L .71901 .01472 L .72022 .01473 L .7215 .01475 L .72272 .01477 L .72382 .0148 L .72653 .01489 L .72908 .015 L .73416 .01531 L .73968 .01576 L .74914 .01685 L .75955 .0185 L .77042 .02071 L .7806 .02325 L .80088 .02967 L .81942 .03713 L .85918 .05836 L .89743 .08568 L .93812 .12242 L .97619 .16418 L Mistroke Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z020003o`000;80 000K0003o`0004d000?o0000/P0001/000?o0000C0000ol0002c00006`000ol0001<0003o`000;<0 000K0003o`0004/000?o0000]00001/00_m;0003o`000;D0000K0003o`0004X000?o0000]@0001/0 00?o0000B@000ol0002f00000P02o`@00_l400;o0`04o`@000?o0000B@000ol0002f000000D0o`00 o`0300;o0`001?l00?l30003o`0000@000?o0000B0000ol0002g000000D0o`00o`080004o`00o`@0 00?o00000`02odP000?o0000^00000050?l00?l02@02o`H000?o00000P000ol000170003o`000;P0 00001@3o003o00P000Co003o0P001?l00?l40003o`0004H000?o0000^@0000800_l:00;o1002o`D0 00?o0000AP000ol0002i00006`000ol000150003o`000;X0000K0003o`0004D000?o0000^P0001/0 0_m50003o`000;/0000K0003o`0004<000?o0000_00001/000?o0000@`000ol0002l00006`000ol0 00120003o`000;d0000K0003o`00048000?o0000_@0001/00_m20003o`000;h0000K0003o`000440 00?o0000_P0001/000?o0000@0000ol0002o00006`000ol000100003o`000;l0000K0003o`0003l0 00?o0000`00001/00_lo0003o`000<40000K0003o`0003h000?o0000`@0001/000?o0000?@000ol0 003200006`000ol0000m0003o`000<80000200;o1002o`@00_l400?o10000ol0000l0003o`000<<0 00001@3o003o00<00_l30004o`00o`@000?o00000`000ol0000l0003o`000<<000001@3o003o00P0 00Co003o0P04o`@00_ll0003o`000<@000001@3o003o00T00_l30003o`3o00D000?o0000>`000ol0 0034000000D0o`00o`080004o`00o`8000?o0?l01@000ol0000j0003o`000@000ol0003600006`02ocX000?o0000aP0001/0 00?o0000>@000ol0003600006`000ol0000h0003o`000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.720586, 0.00394728, 0.000947862}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["Show[p1,p2,p3,p]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.37241 1.97572 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .0106 -18 -4.5 ] [.01131 .0106 0 4.5 ] [.01131 .10939 -24 -4.5 ] [.01131 .10939 0 4.5 ] [.01131 .20817 -18 -4.5 ] [.01131 .20817 0 4.5 ] [.01131 .30696 -24 -4.5 ] [.01131 .30696 0 4.5 ] [.01131 .40575 -18 -4.5 ] [.01131 .40575 0 4.5 ] [.01131 .50453 -24 -4.5 ] [.01131 .50453 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .0106 m .03006 .0106 L s [(0.7)] .01131 .0106 1 0 Mshowa .02381 .10939 m .03006 .10939 L s [(0.75)] .01131 .10939 1 0 Mshowa .02381 .20817 m .03006 .20817 L s [(0.8)] .01131 .20817 1 0 Mshowa .02381 .30696 m .03006 .30696 L s [(0.85)] .01131 .30696 1 0 Mshowa .02381 .40575 m .03006 .40575 L s [(0.9)] .01131 .40575 1 0 Mshowa .02381 .50453 m .03006 .50453 L s [(0.95)] .01131 .50453 1 0 Mshowa .125 Mabswid .02381 .03036 m .02756 .03036 L s .02381 .05012 m .02756 .05012 L s .02381 .06987 m .02756 .06987 L s .02381 .08963 m .02756 .08963 L s .02381 .12914 m .02756 .12914 L s .02381 .1489 m .02756 .1489 L s .02381 .16866 m .02756 .16866 L s .02381 .18842 m .02756 .18842 L s .02381 .22793 m .02756 .22793 L s .02381 .24769 m .02756 .24769 L s .02381 .26745 m .02756 .26745 L s .02381 .2872 m .02756 .2872 L s .02381 .32672 m .02756 .32672 L s .02381 .34647 m .02756 .34647 L s .02381 .36623 m .02756 .36623 L s .02381 .38599 m .02756 .38599 L s .02381 .4255 m .02756 .4255 L s .02381 .44526 m .02756 .44526 L s .02381 .46502 m .02756 .46502 L s .02381 .48478 m .02756 .48478 L s .02381 .52429 m .02756 .52429 L s .02381 .54405 m .02756 .54405 L s .02381 .5638 m .02756 .5638 L s .02381 .58356 m .02756 .58356 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 1 0 r .5 Mabswid .02381 .60332 m .11905 .43707 L .21429 .3072 L .30952 .20586 L .40476 .12826 L .5 .07154 L .59524 .03407 L .69048 .01512 L .78571 .01472 L .88095 .03348 L .97619 .07266 L s 0 0 1 r .02381 .60332 m .07143 .52019 L .11905 .44624 L .16667 .38041 L .21429 .32185 L .2619 .26991 L .30952 .22404 L .35714 .18384 L .40476 .14898 L .45238 .1192 L .5 .09433 L .54762 .07422 L .59524 .05879 L .64286 .048 L .69048 .04186 L .7381 .04041 L .78571 .04373 L .83333 .05192 L .88095 .06514 L .92857 .08358 L .97619 .10744 L s 1 0 0 r .02381 .60332 m .04762 .56176 L .07143 .5225 L .09524 .4854 L .11905 .45035 L .14286 .41723 L .16667 .38594 L .19048 .35641 L .21429 .32854 L .2381 .30227 L .2619 .27754 L .28571 .25429 L .30952 .23247 L .33333 .21204 L .35714 .19296 L .38095 .1752 L .40476 .15871 L .42857 .14349 L .45238 .12949 L .47619 .11672 L .5 .10514 L .52381 .09475 L .54762 .08554 L .57143 .0775 L .59524 .07062 L .61905 .06491 L .64286 .06035 L .66667 .05697 L .69048 .05475 L .71429 .05371 L .7381 .05386 L .7619 .05521 L .78571 .05778 L .80952 .06157 L .83333 .06661 L .85714 .07293 L .88095 .08053 L .90476 .08945 L .92857 .09971 L .95238 .11135 L .97619 .12438 L s 0 g .02381 .60332 m .06244 .53882 L .10458 .47477 L .14415 .42008 L .18221 .37207 L .22272 .32557 L .26171 .285 L .30316 .24613 L .34309 .21262 L .3815 .18388 L .42237 .15692 L .46172 .13438 L .49955 .11582 L .53984 .09933 L .57861 .08663 L .59832 .08136 L .61984 .07652 L .64066 .07274 L .65954 .07009 L .66899 .06904 L .67922 .06811 L .68484 .06769 L .69002 .06736 L .69493 .06711 L .70019 .06688 L .7028 .0668 L .70528 .06673 L .70769 .06667 L .70993 .06663 L .71204 .0666 L .7131 .06659 L .71428 .06658 L .71545 .06658 L .71672 .06657 L .71792 .06657 L .71901 .06657 L .72022 .06658 L .7215 .06659 L .72272 .0666 L .72382 .06661 L .72653 .06666 L .72908 .06671 L .73416 .06687 L .73968 .06709 L .74914 .06764 L .75955 .06845 L .77042 .06956 L .7806 .07082 L .80088 .07402 L .81942 .07774 L Mistroke .85918 .08832 L .89743 .10192 L .93812 .12022 L .97619 .14103 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z0200006`000ol0000/0007g000dR<0o`3?00006`02ob`000GL003B8`020003o`00080 00070004o`00o`P000Co003o10000ol0000K0004g=8Son<0000800;o2P02o`D000?o00006P001=`0 8ooT00006`000ol0000I0005g03B8ol0i00001/00_lI0005g03B8ol0i@0001/000?o000060001=cB 8ooV00006`000ol0000G0004g00SonL0000K0003o`0001H000GL0=8So`3W00006`000ol0000F0004 g=8SonP0000K00;o5P001=cB8ooY00006`000ol0000E0003g2?o0>X0000K0003o`0001@000CLdR?o jP0001/000?o00004`001=cB8oo[00006`000ol0000C0003g2?o0>`0000K00;o4`001=cB8oo/0000 6`000ol0000B0003dR?o0>d0000K0003o`00014000?Ld_l0kP0001/000?o00004@000m;o003^0000 6`000ol0000@0003dR?o0>l0000K0003o`0000l000?Ld_l0l00001/00_l@0003d_l00?00000K0003 o`0000h000?Ld_l0l@0001/000?o00003P000m;o003a00000P02o`@00_l300?o0`03o`D000?o0000 3@000m;o003b000000D0o`00o`0300;o1P000ol000030003o`00008000?o00003@000m;o003b0000 00D0o`00o`0900?o1@000ol0000200;o3@000m;o003c000000D0o`00o`080004o`00o`800ol50003 o`0000/000?Bo`00m00000050?l00?l020001?l00?l20003o`0000D000?o00002`000b?o003d0000 0P02o`X00_l300Co10000ol0000:0003d_l00?D0000K0003o`0000T000?B8ol0mP0001/00_l:0003 8ol00?H0000K0003o`0000P000?Bo`00m`0001/000?o000020000b?o003g00006`000ol000070003 8ol00?P0000K0003o`0000L000?o0000n00001/00_l700038ol00?T0000K0003o`0000D000?Bo`00 nP0001/000?o00001@000b?o003j00006`000ol0000400038ol00?/0000K0003o`0000@000?o0000 n`0001/000?o00000`000b?o003l00006`02o`@000?o00008P02o`@00_l300Co9002o`@00_l400?o 9002o`@00_l400;o9002o`@00_l400;o:P03o`L0000K0003o`00008000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.681111, 0.00394728, 0.00190276}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 7", "Subsubsection"], Cell[TextData[{ "Find approximate values of the solution of the initial value problem\n", StyleBox["\ty'[t] == (t^2 - y[t]^2) Sin[y[t]], y[0] = 1\n", "Input"], "using the Runge-Kutta method with ", StyleBox["h = 0.1", "Input"], ", ", StyleBox["h = 0.05", "Input"], ", and ", StyleBox["h = 0.025", "Input"], ". Compare these results with the output of ", StyleBox["NDSolve[]", "Input"], " (from ", StyleBox["Problem 6", "Subsubsection"], ") by superimposing plots of the data." }], "Text"], Cell["Solution", "Subsubsection"], Cell["Clear[y]", "Input"], Cell[TextData[{ "f[t_,y_] = ", StyleBox["(t^2 - y^2) Sin[y];", "Input"] }], "Input"], Cell["\<\ RKStep[{t_,y_}] := ( k1 = N[f[t,y]]; k2 = N[f[t+0.5*h,y+0.5*h*k1]]; k3 = N[f[t+0.5*h,y+0.5*h*k2]]; k4 = N[f[t+h,y+h*k3]]; N[{t+h,y+(h/6)*(k1+2*k2+2*k3+k4)}] )\ \>", "Input"] }, Open ]], Cell["h = 0.1;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["soln1=NestList[RKStep,{0,1},10]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.1`, 0.924517467294259986`}, {0.2`, 0.864125217985013804`}, { 0.300000000000000044`, 0.816377018901722451`}, {0.4`, 0.779706113125055644`}, {0.5`, 0.753152509747751786`}, {0.6`, 0.736205432485287758`}, {0.7`, 0.728715079522211972`}, { 0.799999999999999911`, 0.730844688807109577`}, {0.9`, 0.74304213782206645`}, {0.999999999999999822`, 0.766012275268258591`}} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ p1 = ListPlot[soln1,PlotJoined->True, PlotStyle->{RGBColor[0,1,0]}]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.56637 2.16969 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .0609 -24 -4.5 ] [.01131 .0609 0 4.5 ] [.01131 .16938 -18 -4.5 ] [.01131 .16938 0 4.5 ] [.01131 .27787 -24 -4.5 ] [.01131 .27787 0 4.5 ] [.01131 .38635 -18 -4.5 ] [.01131 .38635 0 4.5 ] [.01131 .49483 -24 -4.5 ] [.01131 .49483 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .0609 m .03006 .0609 L s [(0.75)] .01131 .0609 1 0 Mshowa .02381 .16938 m .03006 .16938 L s [(0.8)] .01131 .16938 1 0 Mshowa .02381 .27787 m .03006 .27787 L s [(0.85)] .01131 .27787 1 0 Mshowa .02381 .38635 m .03006 .38635 L s [(0.9)] .01131 .38635 1 0 Mshowa .02381 .49483 m .03006 .49483 L s [(0.95)] .01131 .49483 1 0 Mshowa .125 Mabswid .02381 .08259 m .02756 .08259 L s .02381 .10429 m .02756 .10429 L s .02381 .12599 m .02756 .12599 L s .02381 .14768 m .02756 .14768 L s .02381 .19108 m .02756 .19108 L s .02381 .21277 m .02756 .21277 L s .02381 .23447 m .02756 .23447 L s .02381 .25617 m .02756 .25617 L s .02381 .29956 m .02756 .29956 L s .02381 .32126 m .02756 .32126 L s .02381 .34296 m .02756 .34296 L s .02381 .36465 m .02756 .36465 L s .02381 .40805 m .02756 .40805 L s .02381 .42974 m .02756 .42974 L s .02381 .45144 m .02756 .45144 L s .02381 .47314 m .02756 .47314 L s .02381 .51653 m .02756 .51653 L s .02381 .53823 m .02756 .53823 L s .02381 .55993 m .02756 .55993 L s .02381 .58162 m .02756 .58162 L s .02381 .0392 m .02756 .0392 L s .02381 .0175 m .02756 .0175 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 1 0 r .5 Mabswid .02381 .60332 m .11905 .43955 L .21429 .30851 L .30952 .20491 L .40476 .12535 L .5 .06774 L .59524 .03097 L .69048 .01472 L .78571 .01934 L .88095 .0458 L .97619 .09564 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z02g3P0000K0003o`0009h01m`U00CL=00001/0 00?o0000V006g3000m`a00006`000ol0002D00CL>@04g2d0000K0003o`0009400ma100CL:@0001/0 0_n?00?LB003g2H0000K0003o`0008`00]a>00CL8P0001/000?o0000R@03g5@00m`O00006`000ol0 002700;LFP02g1d0000200;o1002o`@000?o00000P03o`D000?o0000Q003g5h00]`K000000D0o`00 o`0300;o10000ol000050003o`00008000?o0000P@03g6<00]`I000000D0o`00o`0:0003o`0000@0 00?o00000P02oh000]aX00;L5`0000050?l00?l02P001?l00003o`D000?o0000O003g6`00]`E0000 00D0o`00o`0;0004o`00o`L000?o0000NP02g7400]`C00000P02o`T01?l200Co10000ol0001h00;L M@000m`0000@00006`000ol0001g0003g00007H00]`@00006`000ol0001e00;LN`02g0h0000K00;o M002g7l00]`<00006`000ol0001b0003g00008400]`:00006`000ol0001`00;LQP02g0P0000K0003 o`0006h00]b:0003g00000D0000K0003o`0006d000?L0000TP0001/000?o0000J`02g9D0000K00;o JP02g9L0000K0003o`0006P000?L0000U`0001/000?o0000IP02g9X0000K0003o`0006@00]bL0000 6`000ol0001S0003g00009`0000K00;oHP02g9l0000K0003o`00060000?L0000W`0001/000?o0000 G`000m`0002P00006`000ol0001N0003g0000:40000K0003o`0005d000?L0000XP0001/000?o0000 F`02g:D0000K00;oF`000m`0002U00006`000ol0001I0003g0000:H0000K0003o`0005P000?L0000 Y`0001/000?o0000E`000m`0002X00002002o`@00_l400;o1@000ol0001E00;LZ`0000L000Co003o 0`02o`<000Co003o10000ol0001D0003g0000:/000070004o`00o`P000Co003o1002oe@000?L0000 [00000L000Co003o2@02o`D000?o0000DP000m`0002]00001`001?l00?l80004o`00o`@000?o0000 D@000m`0002^00002002o`X00_l50003o`00050000?L0000[`0001/000?o0000CP02g;80000K0003 o`0004d000?L0000/P0001/00_m=0003g0000;<0000K0003o`0004/000?L0000]00001/000?o0000 BP000m`0002e00006`000ol000190003g0000;H0000K0003o`0004T000?L0000]P0001/00_m90003 g0000;L0000K0003o`0004L000?L0000^00001/000?o0000AP000m`0002i00006`000ol000150003 g0000;X0000K0003o`0004@000?L0000^`0001/000?o0000@`000m`0002l00006`02od<000?L0000 _@0001/000?o0000@@000m`0002n00006`000ol000100003g0000;l0000K0003o`0003l000?L0000 `00001/000?o0000?P000m`0003100006`000ol0000m0003g0000<80000K00;o?@000m`000330000 6`000ol0000l0003g0000<<0000K0003o`0003/000?L0000a00001/000?o0000>P000m`000350000 0P02o`@00_l400;o0`03o`D000?o0000>@000m`00036000000D0o`00o`0300;o0`001?l00?l50003 o`00008000?o0000>0000m`00037000000D0o`00o`080004o`00o`D000?o00000P02ocP000?L0000 b00000050?l00?l02@02o`<00ol50003o`0003H000?L0000b@0000050?l00?l020001?l00?l20003 o`0000D000?o0000=@000m`0003:00000P02o`X00_l300Co10000ol0000d0003g0000@0000K0003 o`0001/000?L0000i00001/000?o00006P000m`0003U00006`000ol0000I0003g0000>H0000K00;o 6@000m`0003W00006`000ol0000H0003g0000>L0000K0003o`0001L000?L0000j00001/000?o0000 5P000m`0003Y00006`000ol0000F0003g0000>T0000K0003o`0001D000?L0000jP0001/00_lF0003 g0000>X0000K0003o`0001@000?L0000j`0001/000?o00004`000m`0003/00006`000ol0000C0003 g0000>`0000K0003o`00018000?L0000k@0001/000?o00004P000m`0003]00006`02oa8000?L0000 kP0001/000?o000040000m`0003_00006`000ol0000@0003g0000>l0000200;o1002o`<00ol300?o 1@000ol0000?0003g0000?0000001@3o003o00<00_l60003o`0000<000?o00000P000ol0000?0003 g0000?0000001@3o003o00T00ol50003o`0000800_l?0003g0000?4000001@3o003o00P000Co003o 0P03o`D000?o00003P000m`0003a000000D0o`00o`080004o`00o`8000?o00001@000ol0000=0003 g0000?80000200;o2P02o`<01?l40003o`0000`000?L0000l`0001/000?o000030000m`0003c0000 6`000ol0000;0003g0000?@0000K00;o30000m`0003d00006`000ol0000:0003g0000?D0000K0003 o`0000T000?L0000mP0001/000?o00002@000m`0003f00006`000ol000080003g0000?L0000K0003 o`0000P000?L0000m`0001/00_l80003g0000?P0000K0003o`0000L000?L0000n00001/000?o0000 1P000m`0003i00006`000ol000050003g0000?X0000K0003o`0000D000?L0000nP0001/000?o0000 10000m`0003k00006`02o`D000?L0000n`0001/000?o00000`000m`0000R00;o1002o`<01?lT00;o 1002o`@00olT00;o1002o`@00_lT00;o1002o`@00_lZ00?o1`0001/000?o00000`000m`0000Q0004 o`00o`<00_l40003o`0002<000Co003o0`02o`D000?o00008P001?l00?l300;o0`001?l00?lR0004 o`00o`<00_l30004o`00obX000?o00001P0001/000?o00000P000m`0000R0004o`00o`X000?o0000 8P001?l00?l800Co8`001?l00?l80004o`00ob8000Co003o20001?l00?lZ0003o`0000H0000K0005 o`000=`09@001?l00?l;0003o`00024000Co003o20000ol0o`0T0004o`00o`P00olS0004o`00o`T0 0_l[0003o`0000H0000K0005o`000=`09@001?l00?l80004o`00ob<000Co003o20000ol0o`0T0004 o`00o`P000?o00008`001?l00?l80004o`00obX000?o00001P0001/00_l00`3L000V00;o2P02obD0 0_l:00;o9@02o`X00olS00;o2P02obX00_l800006`001?l00=co0080000K0003o`3L0?l00`0001/0 00?og000o`0300006`000ooL003o00<0000D00Oo00?Lool0ool3o`006`000ol000090003o`0000X0 00?o00002P000ol000090003o`0000X000?o00002P000ol000090003o`0000X000?o00002P000ol0 00090003o`0000X000?o00002P000ol000090003o`0000X000?o00002P000ol000090003o`0000X0 00?o00002P000ol000090003o`0000X000?o00001@0001/000?o0000;`000ol0000`0003o`000300 00?o0000;`000ol0000`0003o`0000D0000K0003o`000?l00`0001/000?o0000o`030000o`0Q0000 o`0Q0000o`0Q0000o`0Q0000o`0Q0000o`0Q0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.711071, 0.00394728, 0.00173265}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["h = 0.05;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["soln2=NestList[RKStep,{0,1},20]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.05`, 0.960174125279324641`}, {0.1`, 0.924517198090078196`}, {0.150000000000000017`, 0.892612516779432141`}, {0.2`, 0.86412491826498492`}, {0.25`, 0.838784968872059266`}, {0.3`, 0.816376778988096063`}, {0.35`, 0.796728655751790526`}, {0.399999999999999964`, 0.779705965709839254`}, {0.45`, 0.765205715301370847`}, { 0.499999999999999911`, 0.753152466115976615`}, { 0.549999999999999911`, 0.74349528581259765`}, {0.6`, 0.736205498041423123`}, {0.65`, 0.731275039366136425`}, { 0.700000000000000088`, 0.728715261080030884`}, { 0.750000000000000088`, 0.728556031207112742`}, { 0.800000000000000177`, 0.73084499845512667`}, {0.850000000000000177`, 0.73564687654590779`}, {0.900000000000000177`, 0.743042595122792981`}, {0.950000000000000355`, 0.753128143500808366`}, {1.00000000000000022`, 0.766012907871448245`}} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ p2 = ListPlot[soln2,PlotJoined->True, PlotStyle->{RGBColor[0,0,1]}]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.5651 2.16842 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .06121 -24 -4.5 ] [.01131 .06121 0 4.5 ] [.01131 .16964 -18 -4.5 ] [.01131 .16964 0 4.5 ] [.01131 .27806 -24 -4.5 ] [.01131 .27806 0 4.5 ] [.01131 .38648 -18 -4.5 ] [.01131 .38648 0 4.5 ] [.01131 .4949 -24 -4.5 ] [.01131 .4949 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .06121 m .03006 .06121 L s [(0.75)] .01131 .06121 1 0 Mshowa .02381 .16964 m .03006 .16964 L s [(0.8)] .01131 .16964 1 0 Mshowa .02381 .27806 m .03006 .27806 L s [(0.85)] .01131 .27806 1 0 Mshowa .02381 .38648 m .03006 .38648 L s [(0.9)] .01131 .38648 1 0 Mshowa .02381 .4949 m .03006 .4949 L s [(0.95)] .01131 .4949 1 0 Mshowa .125 Mabswid .02381 .0829 m .02756 .0829 L s .02381 .10458 m .02756 .10458 L s .02381 .12627 m .02756 .12627 L s .02381 .14795 m .02756 .14795 L s .02381 .19132 m .02756 .19132 L s .02381 .213 m .02756 .213 L s .02381 .23469 m .02756 .23469 L s .02381 .25637 m .02756 .25637 L s .02381 .29974 m .02756 .29974 L s .02381 .32142 m .02756 .32142 L s .02381 .34311 m .02756 .34311 L s .02381 .36479 m .02756 .36479 L s .02381 .40816 m .02756 .40816 L s .02381 .42985 m .02756 .42985 L s .02381 .45153 m .02756 .45153 L s .02381 .47321 m .02756 .47321 L s .02381 .51658 m .02756 .51658 L s .02381 .53827 m .02756 .53827 L s .02381 .55995 m .02756 .55995 L s .02381 .58163 m .02756 .58163 L s .02381 .03953 m .02756 .03953 L s .02381 .01785 m .02756 .01785 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .5 Mabswid .02381 .60332 m .07143 .51696 L .11905 .43964 L .16667 .37046 L .21429 .30869 L .2619 .25374 L .30952 .20515 L .35714 .16254 L .40476 .12563 L .45238 .09419 L .5 .06805 L .54762 .04711 L .59524 .0313 L .64286 .02061 L .69048 .01506 L .7381 .01472 L .78571 .01968 L .83333 .03009 L .88095 .04613 L .92857 .068 L .97619 .09594 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z0200?BBP03dRD0000K0003o`0008X00m9@00?B8P0001/000?o0000Q`03dUH00m8O00006`000ol0 002500;BG002dQd0000200;o1002o`@000?o00000P03o`D000?o0000P`02dV000]8K000000D0o`00 o`0300;o10000ol000050003o`00008000?o0000P003dV@00]8I000000D0o`00o`0:0003o`0000@0 00?o00000P02ogl00]9Y00;B5`0000050?l00?l02P001?l00003o`D000?o0000O002dVd00]8E0000 00D0o`00o`0;0004o`00o`L000?o0000NP02dW400]8C00000P02o`T01?l200Co10000ol0001h00;B M@02dQ40000K0003o`0007L000?B0000M`02dPl0000K0003o`0007D00]9l00;B3@0001/00_md00;B P0000m80000:00006`000ol0001b0003dP0008400]8:00006`000ol0001`00;BQP02dPP0000K0003 o`0006h00]::0003dP0000D0000K0003o`0006d000?B0000TP0001/000?o0000J`02dYD0000K00;o JP02dYL0000K0003o`0006P000?B0000U`0001/000?o0000IP02dYX0000K0003o`0006@00]:L0000 6`000ol0001S0003dP0009`0000K00;oHP02dYl0000K0003o`00060000?B0000W`0001/000?o0000 G`000m80002P00006`000ol0001M00;BX`0001/000?o0000G0000m80002S00006`000ol0001K0003 dP000:@0000K00;oF`000m80002U00006`000ol0001H00;BZ00001/000?o0000E`000m80002X0000 6`000ol0001F0003dP000:T0000800;o1002o`@00_l50003o`0005@00]:/00001`001?l00?l300;o 0`001?l00?l40003o`0005<000?B0000[00000L000Co003o20001?l00?l400;oD`000m80002]0000 1`001?l00?l900;o1@000ol0001A0003dP000:h000070004o`00o`P000Co003o10000ol0001@0003 dP000:l0000800;o2P02o`D000?o0000C`000m80002`00006`000ol0001>0003dP000;40000K0003 o`0004d000?B0000/P0001/00_m=0003dP000;<0000K0003o`0004/000?B0000]00001/000?o0000 BP000m80002e00006`000ol000190003dP000;H0000K0003o`0004P000?B0000]`0001/00_m80003 dP000;P0000K0003o`0004H000?B0000^@0001/000?o0000A@000m80002j00006`000ol000140003 dP000;/0000K0003o`0004<000?B0000_00001/000?o0000@P000m80002m00006`02od8000?B0000 _P0001/000?o0000@0000m80002o00006`000ol0000o0003dP000<00000K0003o`0003h000?B0000 `@0001/000?o0000?@000m80003200006`000ol0000l0003dP000<<0000K00;o?@000m8000330000 6`000ol0000k0003dP000<@0000K0003o`0003X000?B0000a@0001/000?o0000>@000m8000360000 0P02o`@00_l400;o0`03o`D000?o0000>0000m800037000000D0o`00o`0300;o0`001?l00?l50003 o`00008000?o0000=`000m800038000000D0o`00o`080004o`00o`D000?o00000P02ocL000?B0000 b@0000050?l00?l02@02o`<00ol50003o`0003D000?B0000bP0000050?l00?l020001?l00?l20003 o`0000D000?o0000=@000m80003:00000P02o`X00_l300Co10000ol0000d0003dP000@0000K0003 o`0001X000?B0000i@0001/000?o00006P000m80003U00006`000ol0000I0003dP000>H0000K00;o 6@000m80003W00006`000ol0000H0003dP000>L0000K0003o`0001L000?B0000j00001/000?o0000 5P000m80003Y00006`000ol0000F0003dP000>T0000K0003o`0001D000?B0000jP0001/00_lE0003 dP000>/0000K0003o`0001@000?B0000j`0001/000?o00004`000m80003/00006`000ol0000B0003 dP000>d0000K0003o`00018000?B0000k@0001/000?o00004@000m80003^00006`02oa4000?B0000 k`0001/000?o000040000m80003_00006`000ol0000?0003dP000?00000200;o1002o`<00ol300?o 1@000ol0000?0003dP000?0000001@3o003o00<00_l60003o`0000<000?o00000P000ol0000>0003 dP000?4000001@3o003o00T00ol50003o`0000800_l>0003dP000?8000001@3o003o00P000Co003o 0P03o`D000?o00003@000m80003b000000D0o`00o`080004o`00o`8000?o00001@000ol0000<0003 dP000?<0000200;o2P02o`<01?l40003o`0000/000?B0000m00001/000?o00002`000m80003d0000 6`000ol0000:0003dP000?D0000K00;o2P000m80003f00006`000ol000090003dP000?H0000K0003 o`0000P000?B0000m`0001/000?o000020000m80003g00006`000ol000070003dP000?P0000K0003 o`0000L000?B0000n00001/00_l70003dP000?T0000K0003o`0000H000?B0000n@0001/000?o0000 1@000m80003j00006`000ol000050003dP000?X0000K0003o`0000@000?B0000n`0001/000?o0000 10000m80003k00006`02o`@000?B0000o00001/000?o00000P000m80000S00;o1002o`<01?lT00;o 1002o`@00olT00;o1002o`@00_lT00;o1002o`@00_lZ00?o1`0001/000?o00000P000m80000R0004 o`00o`<00_l40003o`0002<000Co003o0`02o`D000?o00008P001?l00?l300;o0`001?l00?lR0004 o`00o`<00_l30004o`00obX000?o00001P0001/000Go0000dP0U0004o`00o`X000?o00008P001?l0 0?l800Co8`001?l00?l80004o`00ob8000Co003o20001?l00?lZ0003o`0000H0000K0005o`000=80 9@001?l00?l;0003o`00024000Co003o20000ol0o`0T0004o`00o`P00olS0004o`00o`T00_l[0003 o`0000H0000K0004o`00dRH000Co003o20001?l00?lS0004o`00o`P000?o0?l090001?l00?l80003 o`0002<000Co003o20001?l00?lZ0003o`0000H0000K00;o00<0dP009P02o`X00_lU00;o2P02obD0 0_l:00?o8`02o`X00_lZ00;o200001/000?o0=80o`0300006`000ol0dP3o00<0000K0003om800?l0 0`0001/000?odP00o`0300005007o`03d_oo0?oo0ol001/000?o00002@000ol0000:0003o`0000X0 00?o00002@000ol0000:0003o`0000X000?o00002@000ol0000:0003o`0000X000?o00002@000ol0 000:0003o`0000X000?o00002@000ol0000:0003o`0000X000?o00002@000ol0000:0003o`0000X0 00?o00002@000ol0000:0003o`0000D0000K0003o`0002l000?o0000<0000ol0000`0003o`0002l0 00?o0000<0000ol0000500006`000ol0003o00<0000K0003o`000?l00`000?l08@000?l08@000?l0 8@000?l08@000?l08@000?l08@000001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.710901, 0.00394728, 0.00173367}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["h = 0.025;", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["soln3=NestList[RKStep,{0,1},40]", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{0, 1}, {0.025`, 0.979536186805844089`}, {0.05`, 0.9601741130222182`}, {0.0750000000000000088`, 0.941852696126110089`}, {0.1`, 0.924517179459673776`}, {0.125`, 0.908118460015493766`}, {0.15`, 0.892612495463491839`}, {0.175`, 0.877959782186302817`}, {0.199999999999999982`, 0.864124896577481571`}, {0.225`, 0.851076092474075629`}, { 0.249999999999999955`, 0.838784948260743412`}, { 0.274999999999999955`, 0.827226057846965545`}, {0.3`, 0.816376760352759767`}, {0.325`, 0.806216903925837691`}, { 0.350000000000000044`, 0.796728639647019498`}, { 0.375000000000000044`, 0.787896241958892407`}, { 0.400000000000000088`, 0.779705952476306052`}, { 0.425000000000000088`, 0.772145844409213566`}, { 0.450000000000000088`, 0.765205705152359527`}, { 0.475000000000000177`, 0.758876934876431707`}, { 0.500000000000000088`, 0.753152459195553092`}, { 0.525000000000000177`, 0.748026654190146356`}, { 0.550000000000000177`, 0.743495282235655885`}, { 0.575000000000000177`, 0.73955543722935424`}, {0.600000000000000177`, 0.736205497922064111`}, {0.625000000000000177`, 0.733445088151259216`}, {0.650000000000000266`, 0.731275042838476529`}, {0.675000000000000266`, 0.729697378658746398`}, {0.700000000000000266`, 0.728715268314025355`}, {0.725000000000000355`, 0.728333017347399902`}, {0.750000000000000355`, 0.728556042420947402`}, {0.775000000000000355`, 0.729390849948397512`}, {0.800000000000000355`, 0.730845013924951558`}, {0.825000000000000355`, 0.732927151731769477`}, {0.850000000000000355`, 0.735646896612987433`}, {0.875000000000000355`, 0.739014865430392831`}, {0.900000000000000532`, 0.743042620197397418`}, {0.925000000000000532`, 0.747742621782895255`}, {0.950000000000000532`, 0.753128174061246991`}, {0.975000000000000532`, 0.759213356672614025`}, {1.00000000000000044`, 0.766012944455460775`}} \)], "Output"] }, Open ]], Cell[TextData[{ "The value of ", StyleBox["y[1]", "Input"], " for each of these approximations and for the output of ", StyleBox["NDSolve[]", "Input"], " is within ", StyleBox["\[PlusMinus]0.000001", "Input"], " of ", StyleBox["0.766013", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["\<\ p3 = ListPlot[soln3,PlotJoined->True, PlotStyle->{RGBColor[1,0,0]}]\ \>", "Input", CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.56332 2.16664 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .06166 -24 -4.5 ] [.01131 .06166 0 4.5 ] [.01131 .16999 -18 -4.5 ] [.01131 .16999 0 4.5 ] [.01131 .27832 -24 -4.5 ] [.01131 .27832 0 4.5 ] [.01131 .38666 -18 -4.5 ] [.01131 .38666 0 4.5 ] [.01131 .49499 -24 -4.5 ] [.01131 .49499 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .06166 m .03006 .06166 L s [(0.75)] .01131 .06166 1 0 Mshowa .02381 .16999 m .03006 .16999 L s [(0.8)] .01131 .16999 1 0 Mshowa .02381 .27832 m .03006 .27832 L s [(0.85)] .01131 .27832 1 0 Mshowa .02381 .38666 m .03006 .38666 L s [(0.9)] .01131 .38666 1 0 Mshowa .02381 .49499 m .03006 .49499 L s [(0.95)] .01131 .49499 1 0 Mshowa .125 Mabswid .02381 .08333 m .02756 .08333 L s .02381 .10499 m .02756 .10499 L s .02381 .12666 m .02756 .12666 L s .02381 .14833 m .02756 .14833 L s .02381 .19166 m .02756 .19166 L s .02381 .21332 m .02756 .21332 L s .02381 .23499 m .02756 .23499 L s .02381 .25666 m .02756 .25666 L s .02381 .29999 m .02756 .29999 L s .02381 .32166 m .02756 .32166 L s .02381 .34332 m .02756 .34332 L s .02381 .36499 m .02756 .36499 L s .02381 .40832 m .02756 .40832 L s .02381 .42999 m .02756 .42999 L s .02381 .45165 m .02756 .45165 L s .02381 .47332 m .02756 .47332 L s .02381 .51665 m .02756 .51665 L s .02381 .53832 m .02756 .53832 L s .02381 .55999 m .02756 .55999 L s .02381 .58165 m .02756 .58165 L s .02381 .03999 m .02756 .03999 L s .02381 .01833 m .02756 .01833 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .5 Mabswid .02381 .60332 m .04762 .55898 L .07143 .51703 L .09524 .47733 L .11905 .43978 L .14286 .40424 L .16667 .37065 L .19048 .3389 L .21429 .30893 L .2381 .28065 L .2619 .25402 L .28571 .22898 L .30952 .20547 L .33333 .18346 L .35714 .1629 L .38095 .14377 L .40476 .12602 L .42857 .10964 L .45238 .0946 L .47619 .08089 L .5 .06849 L .52381 .05738 L .54762 .04757 L .57143 .03903 L .59524 .03177 L .61905 .02579 L .64286 .02109 L .66667 .01767 L .69048 .01554 L .71429 .01472 L .7381 .0152 L .7619 .01701 L .78571 .02016 L .80952 .02467 L .83333 .03056 L .85714 .03786 L .88095 .04659 L .90476 .05677 L .92857 .06844 L .95238 .08162 L .97619 .09635 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z0200:00038`0000D0000K0003o`0006`00R>D00006`000ol0001[00038`0009@0000K00;o JP028iL0000K0003o`0006L00R>I00006`000ol0001V00038`0009T0000K0003o`0006@00R>L0000 6`000ol0001S00038`0009`0000K0003o`0006400R>O00006`02of4000@000b<000360000 0P02o`@00_l400;o0`03o`D000?o0000>0000b<00037000000D0o`00o`0300;o0`001?l00?l50003 o`00008000?o0000=`000b<00038000000D0o`00o`080004o`00o`D000?o00000P02ocL000D0000K0003 o`0001X000H0000K00;o 6@000b<0003W00006`000ol0000G00038`000>P0000K0003o`0001L000T0000K0003o`0001D000/0000K0003o`0001@000`0000K0003o`00018000l0000200;o1002o`<00ol300?o 1@000ol0000?00038`000?0000001@3o003o00<00_l60003o`0000<000?o00000P000ol0000>0003 8`000?4000001@3o003o00T00ol50003o`0000800_l?00038`000?4000001@3o003o00P000Co003o 0P03o`D000?o00003@000b<0003b000000D0o`00o`080004o`00o`8000?o00001@000ol0000<0003 8`000?<0000200;o2P02o`<01?l40003o`0000/000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.710664, 0.00394728, 0.00173509}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["Show[p1,p2,p3,p]", "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -1.56328 2.1666 [ [.21429 .59082 -9 -9 ] [.21429 .59082 9 0 ] [.40476 .59082 -9 -9 ] [.40476 .59082 9 0 ] [.59524 .59082 -9 -9 ] [.59524 .59082 9 0 ] [.78571 .59082 -9 -9 ] [.78571 .59082 9 0 ] [.97619 .59082 -3 -9 ] [.97619 .59082 3 0 ] [.01131 .06167 -24 -4.5 ] [.01131 .06167 0 4.5 ] [.01131 .17 -18 -4.5 ] [.01131 .17 0 4.5 ] [.01131 .27833 -24 -4.5 ] [.01131 .27833 0 4.5 ] [.01131 .38666 -18 -4.5 ] [.01131 .38666 0 4.5 ] [.01131 .49499 -24 -4.5 ] [.01131 .49499 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .60332 m .21429 .60957 L s [(0.2)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(0.4)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(0.6)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(0.8)] .78571 .59082 0 1 Mshowa .97619 .60332 m .97619 .60957 L s [(1)] .97619 .59082 0 1 Mshowa .125 Mabswid .07143 .60332 m .07143 .60707 L s .11905 .60332 m .11905 .60707 L s .16667 .60332 m .16667 .60707 L s .2619 .60332 m .2619 .60707 L s .30952 .60332 m .30952 .60707 L s .35714 .60332 m .35714 .60707 L s .45238 .60332 m .45238 .60707 L s .5 .60332 m .5 .60707 L s .54762 .60332 m .54762 .60707 L s .64286 .60332 m .64286 .60707 L s .69048 .60332 m .69048 .60707 L s .7381 .60332 m .7381 .60707 L s .83333 .60332 m .83333 .60707 L s .88095 .60332 m .88095 .60707 L s .92857 .60332 m .92857 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .02381 .06167 m .03006 .06167 L s [(0.75)] .01131 .06167 1 0 Mshowa .02381 .17 m .03006 .17 L s [(0.8)] .01131 .17 1 0 Mshowa .02381 .27833 m .03006 .27833 L s [(0.85)] .01131 .27833 1 0 Mshowa .02381 .38666 m .03006 .38666 L s [(0.9)] .01131 .38666 1 0 Mshowa .02381 .49499 m .03006 .49499 L s [(0.95)] .01131 .49499 1 0 Mshowa .125 Mabswid .02381 .08333 m .02756 .08333 L s .02381 .105 m .02756 .105 L s .02381 .12667 m .02756 .12667 L s .02381 .14833 m .02756 .14833 L s .02381 .19166 m .02756 .19166 L s .02381 .21333 m .02756 .21333 L s .02381 .235 m .02756 .235 L s .02381 .25666 m .02756 .25666 L s .02381 .29999 m .02756 .29999 L s .02381 .32166 m .02756 .32166 L s .02381 .34333 m .02756 .34333 L s .02381 .36499 m .02756 .36499 L s .02381 .40832 m .02756 .40832 L s .02381 .42999 m .02756 .42999 L s .02381 .45166 m .02756 .45166 L s .02381 .47332 m .02756 .47332 L s .02381 .51665 m .02756 .51665 L s .02381 .53832 m .02756 .53832 L s .02381 .55999 m .02756 .55999 L s .02381 .58165 m .02756 .58165 L s .02381 .04 m .02756 .04 L s .02381 .01834 m .02756 .01834 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 1 0 r .5 Mabswid .02381 .60332 m .11905 .43978 L .21429 .30893 L .30952 .20548 L .40476 .12603 L .5 .0685 L .59524 .03178 L .69048 .01555 L .78571 .02017 L .88095 .04659 L .97619 .09636 L s 0 0 1 r .02381 .60332 m .07143 .51703 L .11905 .43978 L .16667 .37065 L .21429 .30893 L .2619 .25403 L .30952 .20548 L .35714 .16291 L .40476 .12603 L .45238 .09461 L .5 .0685 L .54762 .04758 L .59524 .03178 L .64286 .0211 L .69048 .01555 L .7381 .01521 L .78571 .02017 L .83333 .03057 L .88095 .04659 L .92857 .06845 L .97619 .09636 L s 1 0 0 r .02381 .60332 m .04762 .55898 L .07143 .51703 L .09524 .47734 L .11905 .43978 L .14286 .40425 L .16667 .37065 L .19048 .33891 L .21429 .30893 L .2381 .28066 L .2619 .25403 L .28571 .22899 L .30952 .20548 L .33333 .18347 L .35714 .16291 L .38095 .14377 L .40476 .12603 L .42857 .10965 L .45238 .09461 L .47619 .0809 L .5 .0685 L .52381 .05739 L .54762 .04758 L .57143 .03904 L .59524 .03178 L .61905 .0258 L .64286 .0211 L .66667 .01768 L .69048 .01555 L .71429 .01472 L .7381 .01521 L .7619 .01702 L .78571 .02017 L .80952 .02468 L .83333 .03057 L .85714 .03787 L .88095 .04659 L .90476 .05678 L .92857 .06845 L .95238 .08163 L .97619 .09636 L s 0 g .02381 .60332 m .06244 .53259 L .10458 .46236 L .14415 .40238 L .18221 .34973 L .22272 .29873 L .26171 .25424 L .30316 .21162 L .34309 .17488 L .3815 .14335 L .42237 .11379 L .46172 .08908 L .49955 .06872 L .53984 .05064 L .57861 .03672 L .59832 .03094 L .61984 .02563 L .64066 .02148 L .65954 .01857 L .66899 .01742 L .67922 .0164 L .68484 .01594 L .69002 .01558 L .69493 .0153 L .70019 .01506 L .7028 .01496 L .70528 .01489 L .70769 .01482 L .70993 .01478 L .71204 .01475 L .7131 .01474 L .71428 .01473 L .71545 .01472 L .71672 .01472 L .71792 .01472 L .71901 .01472 L .72022 .01472 L .7215 .01473 L .72272 .01475 L .72382 .01476 L .72653 .01481 L .72908 .01487 L .73416 .01504 L .73968 .01529 L .74914 .01588 L .75955 .01678 L .77042 .01799 L .7806 .01938 L .80088 .02288 L .81942 .02696 L Mistroke .85918 .03856 L .89743 .05348 L .93812 .07355 L .97619 .09636 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggYjN[Vi^OShn03ooeGooj[ooonIgYjN[Vi^OShn03o oclo?ginO[fm_Oclo5D00:X00?l00?l0VOl0I_l0CXf=SLcCXf=SLcII03=VoalO7eiN GYfMWMcLg02Z020004g?ooob80000K0003 o`0008L000DSoooog01D00;o0000006`000ol0001d0003 8ooo07/00R<00ol0000;00006`02og@00_mo00038ooo00/0000K0003o`00074000P000ooL003500006`000ol0000i0003om`0000006`000ol0000`00038ol000000070004o`00o`P000Co003o10000ol0 000O0003om`00>0000070004o`00o`P000Co003o10000ol0000N00038ol00>40000800;o2P02o`D0 00?o00007@000b?o003R00006`000ol0000L00038ol00><0000K00;o7@000b?o003S00006`000ol0 000K00038ol00>@0000K0003o`0001X000H0000K0003o`0001T000?o0000iP0001/00_lI00038ol00>L0000K0003o`0001L0 00T0000K0003o`0001H0 00?o0000j@0001/000?o00005@000ol0003Z00006`02oaD000?og000j`0001/000?o000050000ol0 003[00006`000ol0000C0003o`000>`0000K0003o`00018000?o8`00k@0001/000?o00004P000ol0 003]00006`000ol0000A0003om`00>h0000K00;o4@000m;o003_00006`000ol0000@0003o`000>l0 000K0003o`0000l000?o8`00l00000800_l400;o0`03o`<00ol50003o`0000l000?o0000l0000005 0?l00?l00`02o`H000?o00000`000ol000020003o`0000h000?og000l@0000050?l00?l02@03o`D0 00?o00000P02o`h000?Bo`00lP0000050?l00?l020001?l00?l200?o1@000ol0000=0003om`00?80 00001@3o003o00P000Co003o0P000ol000050003o`0000`000?og000l`0000800_l:00;o0`04o`@0 00?o00002`000b?o003d00006`000ol0000;0003om`00?@0000K0003o`0000X000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.107864, 0.710659, 0.00394728, 0.00173512}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["The plots are almost identical as well.", "Text"] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, CellGrouping->Manual, WindowSize->{651, 647}, WindowMargins->{{Automatic, 47}, {Automatic, 4}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, MacintoshSystemPageSetup->"\<\ 00<0004/0B`000003il;m?o" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 101, 3, 44, "Subsection"], Cell[CellGroupData[{ Cell[1835, 56, 34, 0, 41, "Subsubsection"], Cell[1872, 58, 311, 8, 113, "Text"], Cell[2186, 68, 33, 0, 41, "Subsubsection"], Cell[2222, 70, 112, 4, 27, "Input"] }, Open ]], Cell[2349, 77, 35, 0, 27, "Input"], Cell[2387, 79, 115, 2, 42, "Input"], Cell[CellGroupData[{ Cell[2527, 85, 64, 0, 27, "Input"], Cell[2594, 87, 46, 1, 26, "Output"] }, Open ]], Cell[2655, 91, 253, 6, 87, "Input"], Cell[CellGroupData[{ Cell[2933, 101, 43, 0, 27, "Input"], Cell[2979, 103, 50, 1, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3066, 109, 34, 0, 41, "Subsubsection"], Cell[3103, 111, 326, 12, 47, "Text"], Cell[3432, 125, 33, 0, 41, "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell[3502, 130, 52, 0, 27, "Input"], Cell[3557, 132, 306, 6, 62, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3900, 143, 28, 0, 27, "Input"], Cell[3931, 145, 35, 1, 26, "Output"] }, Open ]], Cell[3981, 149, 342, 7, 64, "Text"], Cell[CellGroupData[{ Cell[4348, 160, 91, 1, 27, "Input"], Cell[4442, 163, 90, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4569, 169, 190, 5, 42, "Input"], Cell[4762, 176, 140, 3, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4939, 184, 94, 3, 27, "Input"], Cell[5036, 189, 113, 2, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5186, 196, 90, 3, 27, "Input"], Cell[5279, 201, 89, 1, 28, "Output"] }, Open ]], Cell[5383, 205, 60, 0, 30, "Text"], Cell[5446, 207, 69, 0, 27, "Input"], Cell[CellGroupData[{ Cell[5540, 211, 34, 0, 41, "Subsubsection"], Cell[5577, 213, 176, 5, 31, "Text"], Cell[5756, 220, 33, 0, 41, "Subsubsection"], Cell[CellGroupData[{ Cell[5814, 224, 36, 0, 27, "Input"], Cell[5853, 226, 644, 10, 230, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[6546, 242, 39, 0, 27, "Input"], Cell[6588, 244, 535, 9, 185, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[7160, 258, 45, 0, 27, "Input"], Cell[7208, 260, 10228, 352, 296, 3296, 262, "GraphicsData", "PostScript", "Graphics"], Cell[17439, 614, 130, 3, 22, "Output"] }, Open ]], Cell[17584, 620, 122, 4, 25, "Text"], Cell[CellGroupData[{ Cell[17731, 628, 31, 0, 22, "Input"], Cell[17765, 630, 37, 1, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[17839, 636, 73, 0, 22, "Input"], Cell[17915, 638, 558, 8, 103, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[18510, 651, 34, 0, 33, "Subsubsection"], Cell[18547, 653, 548, 11, 77, "Text"], Cell[19098, 666, 33, 0, 33, "Subsubsection"], Cell[CellGroupData[{ Cell[19156, 670, 58, 0, 22, "Input"], Cell[19217, 672, 149, 2, 22, "Output"] }, Open ]] }, Open ]], Cell[19393, 678, 60, 0, 25, "Text"], Cell[19456, 680, 124, 3, 30, "Input"], Cell[CellGroupData[{ Cell[19605, 687, 70, 0, 22, "Input"], Cell[19678, 689, 19145, 821, 185, 10694, 712, "GraphicsData", "PostScript", "Graphics"], Cell[38826, 1512, 130, 3, 22, "Output"] }, Open ]], Cell[38971, 1518, 102, 4, 25, "Text"], Cell[CellGroupData[{ Cell[39098, 1526, 56, 0, 22, "Input"], Cell[39157, 1528, 158, 2, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[39352, 1535, 71, 0, 22, "Input"], Cell[39426, 1537, 176, 3, 22, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[39639, 1545, 28, 0, 22, "Input"], Cell[39670, 1547, 148, 2, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[39855, 1554, 35, 0, 22, "Input"], Cell[39893, 1556, 195, 3, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40125, 1564, 35, 0, 27, "Input"], Cell[40163, 1566, 13126, 481, 186, 5389, 381, "GraphicsData", "PostScript", "Graphics"], Cell[53292, 2049, 130, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[53459, 2057, 220, 7, 42, "Input"], Cell[53682, 2066, 347, 7, 136, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[54066, 2078, 47, 0, 27, "Input"], Cell[54116, 2080, 13126, 481, 186, 5389, 381, "GraphicsData", "PostScript", "Graphics"], Cell[67245, 2563, 130, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67412, 2571, 34, 0, 41, "Subsubsection"], Cell[67449, 2573, 699, 15, 114, "Text"], Cell[68151, 2590, 33, 0, 41, "Subsubsection"], Cell[CellGroupData[{ Cell[68209, 2594, 55, 0, 27, "Input"], Cell[68267, 2596, 124, 2, 26, "Output"] }, Open ]] }, Open ]], Cell[68418, 2602, 68, 0, 30, "Text"], Cell[68489, 2604, 75, 0, 27, "Input"], Cell[68567, 2606, 89, 3, 30, "Text"], Cell[68659, 2611, 137, 4, 27, "Input"], Cell[68799, 2617, 323, 7, 47, "Text"], Cell[69125, 2626, 72, 0, 27, "Input"], Cell[CellGroupData[{ Cell[69222, 2630, 131, 5, 27, "Input"], Cell[69356, 2637, 218, 3, 57, "Output"] }, Open ]], Cell[69589, 2643, 58, 0, 30, "Text"], Cell[69650, 2645, 101, 3, 42, "Input"], Cell[CellGroupData[{ Cell[69776, 2652, 34, 0, 27, "Input"], Cell[69813, 2654, 123, 2, 42, "Output"] }, Open ]], Cell[69951, 2659, 88, 3, 30, "Text"], Cell[CellGroupData[{ Cell[70064, 2666, 52, 0, 27, "Input"], Cell[70119, 2668, 202, 4, 76, "Output"] }, Open ]], Cell[70336, 2675, 45, 0, 30, "Text"], Cell[CellGroupData[{ Cell[70406, 2679, 36, 0, 27, "Input"], Cell[70445, 2681, 58, 1, 27, "Output"] }, Open ]], Cell[70518, 2685, 37, 0, 30, "Text"], Cell[CellGroupData[{ Cell[70580, 2689, 75, 0, 27, "Input"], Cell[70658, 2691, 177, 3, 76, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[70872, 2699, 48, 0, 27, "Input"], Cell[70923, 2701, 223, 4, 76, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[71183, 2710, 49, 0, 27, "Input"], Cell[71235, 2712, 326, 6, 110, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[71598, 2723, 92, 3, 42, "Input"], Cell[71693, 2728, 37937, 1845, 186, 25971, 1693, "GraphicsData", "PostScript", "Graphics"], Cell[109633, 4575, 130, 3, 26, "Output"] }, Open ]], Cell[109778, 4581, 131, 4, 31, "Text"], Cell[CellGroupData[{ Cell[109934, 4589, 165, 2, 60, "Input"], Cell[110102, 4593, 92, 1, 42, "Output"] }, Open ]], Cell[110209, 4597, 352, 11, 65, "Text"], Cell[CellGroupData[{ Cell[110586, 4612, 34, 0, 41, "Subsubsection"], Cell[110623, 4614, 491, 14, 81, "Text"], Cell[111117, 4630, 33, 0, 41, "Subsubsection"], Cell[111153, 4632, 88, 3, 27, "Input"], Cell[111244, 4637, 56, 0, 27, "Input"] }, Open ]], Cell[111315, 4640, 108, 2, 27, "Input"], Cell[111426, 4644, 157, 6, 31, "Text"], Cell[CellGroupData[{ Cell[111608, 4654, 131, 2, 27, "Input"], Cell[111742, 4658, 454, 7, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[112233, 4670, 178, 5, 42, "Input"], Cell[112414, 4677, 10570, 337, 186, 3419, 244, "GraphicsData", "PostScript", "Graphics"], Cell[122987, 5016, 130, 3, 26, "Output"] }, Closed]], Cell[123132, 5022, 109, 2, 24, "Input"], Cell[123244, 5026, 53, 0, 30, "Text"], Cell[CellGroupData[{ Cell[123322, 5030, 131, 2, 27, "Input"], Cell[123456, 5034, 984, 16, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[124477, 5055, 178, 5, 42, "Input"], Cell[124658, 5062, 10500, 339, 186, 3459, 248, "GraphicsData", "PostScript", "Graphics"], Cell[135161, 5403, 130, 3, 26, "Output"] }, Closed]], Cell[135306, 5409, 110, 2, 24, "Input"], Cell[CellGroupData[{ Cell[135441, 5415, 131, 2, 27, "Input"], Cell[135575, 5419, 2130, 36, 161, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[137742, 5460, 178, 5, 42, "Input"], Cell[137923, 5467, 10820, 357, 186, 3746, 265, "GraphicsData", "PostScript", "Graphics"], Cell[148746, 5826, 130, 3, 26, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[148913, 5834, 227, 6, 39, "Input"], Cell[149143, 5842, 271, 7, 26, "Output"] }, Open ]], Cell[149429, 5852, 126, 2, 22, "Input"], Cell[CellGroupData[{ Cell[149580, 5858, 23, 0, 27, "Input"], Cell[149606, 5860, 55, 1, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[149698, 5866, 122, 2, 27, "Input"], Cell[149823, 5870, 11991, 390, 186, 4159, 289, "GraphicsData", "PostScript", "Graphics"], Cell[161817, 6262, 130, 3, 26, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[161984, 6270, 33, 0, 24, "Input"], Cell[162020, 6272, 13875, 478, 186, 5286, 367, "GraphicsData", "PostScript", "Graphics"], Cell[175898, 6752, 130, 3, 26, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[176065, 6760, 34, 0, 41, "Subsubsection"], Cell[176102, 6762, 514, 14, 81, "Text"], Cell[176619, 6778, 33, 0, 41, "Subsubsection"], Cell[176655, 6780, 25, 0, 27, "Input"], Cell[176683, 6782, 88, 3, 27, "Input"], Cell[176774, 6787, 191, 7, 102, "Input"] }, Open ]], Cell[176980, 6797, 108, 2, 27, "Input"], Cell[CellGroupData[{ Cell[177113, 6803, 131, 2, 27, "Input"], Cell[177247, 6807, 454, 7, 56, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[177738, 6819, 180, 5, 42, "Input"], Cell[177921, 6826, 10329, 324, 186, 3239, 232, "GraphicsData", "PostScript", "Graphics"], Cell[188253, 7152, 130, 3, 26, "Output"] }, Closed]], Cell[188398, 7158, 109, 2, 24, "Input"], Cell[CellGroupData[{ Cell[188532, 7164, 131, 2, 27, "Input"], Cell[188666, 7168, 982, 16, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[189685, 7189, 180, 5, 42, "Input"], Cell[189868, 7196, 10474, 334, 186, 3392, 242, "GraphicsData", "PostScript", "Graphics"], Cell[200345, 7532, 130, 3, 26, "Output"] }, Closed]], Cell[200490, 7538, 110, 2, 24, "Input"], Cell[CellGroupData[{ Cell[200625, 7544, 131, 2, 27, "Input"], Cell[200759, 7548, 2131, 36, 161, "Output"] }, Open ]], Cell[202905, 7587, 282, 10, 48, "Text"], Cell[CellGroupData[{ Cell[203212, 7601, 180, 5, 42, "Input"], Cell[203395, 7608, 10811, 354, 186, 3721, 262, "GraphicsData", "PostScript", "Graphics"], Cell[214209, 7964, 130, 3, 26, "Output"] }, Closed]], Cell[CellGroupData[{ Cell[214376, 7972, 33, 0, 24, "Input"], Cell[214412, 7974, 12365, 449, 186, 5089, 355, "GraphicsData", "PostScript", "Graphics"], Cell[226780, 8425, 130, 3, 22, "Output"] }, Open ]], Cell[226925, 8431, 55, 0, 25, "Text"] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)