(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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[ 26197, 774]*) (*NotebookOutlinePosition[ 26826, 796]*) (* CellTagsIndexPosition[ 26782, 792]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Problem 18, p.178", "Subsection"], Cell[TextData[{ "Solve ", Cell[BoxData[ \(TraditionalForm\`y''\ + \ 2 y'\ + \ 5 y\ = \ 4 \( e\^\(-t\)\) \(cos(2 t)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y(0) = 1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`y' \((0)\)\ = \ 0\)]], "." }], "Text"], Cell[TextData[StyleBox["Solution", FontSlant->"Italic"]], "Text"], Cell["Characteristic polynomial:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[r^2\ + \ 2 r\ + \ 5\ \[Equal] \ 0]\)], "Input"], Cell[BoxData[ \({{r \[Rule] \(-1\) - 2\ \[ImaginaryI]}, {r \[Rule] \(-1\) + 2\ \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(yp[t_]\ = t\ \(\[ExponentialE]\^\(-t\)\) \((A\ Cos[2 t]\ + \ B\ Sin[2 t])\);\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(yp'\)[t] // Simplify\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-t\)\ \((\((A - A\ t + 2\ B\ t)\)\ Cos[ 2\ t] + \((B - 2\ A\ t - B\ t)\)\ Sin[2\ t])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(yp''\)[t] // Simplify\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(-t\)\ \((\(-\((4\ B\ \((\(-1\) + t)\) + A\ \((2 + 3\ t)\))\)\)\ Cos[ 2\ t] + \((4\ A\ \((\(-1\) + t)\) - B\ \((2 + 3\ t)\))\)\ Sin[ 2\ t])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(yp''\)[t]\ + \ 2 \( yp'\)[t]\ + \ 5 yp[t] // Simplify\)], "Input"], Cell[BoxData[ \(4\ \[ExponentialE]\^\(-t\)\ \((B\ Cos[2\ t] - A\ Sin[2\ t])\)\)], "Output"] }, Open ]], Cell[TextData[{ "This equals ", Cell[BoxData[ \(TraditionalForm\`4 \( e\^\(-t\)\) \(cos(2 t)\)\)]], " when ", Cell[BoxData[ \(TraditionalForm\`A\ = \ 0\)]], "and ", Cell[BoxData[ \(TraditionalForm\`B\ = \ 1\)]], ", so a particular solution is ", Cell[BoxData[ \(TraditionalForm\`y\_p = \ \(\(t\)\(\ \)\(e\^\(-t\)\) \(sin( 2 t)\)\(\ \)\)\)]], " and the general solution is:" }], "Text"], Cell[BoxData[ \(\(y[ t_]\ = \ \[ExponentialE]\^\(-t\)\ \((c1\ Cos[2 t]\ + \ c2\ Sin[2 t])\) + t\ \(\[ExponentialE]\^\(-t\)\) Sin[2 t];\)\)], "Input"], Cell["Initial conditions:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(y[0] \[Equal] 1\)], "Input"], Cell[BoxData[ \(c1 == 1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(y'\)[0] \[Equal] 0\)], "Input"], Cell[BoxData[ \(\(-c1\) + 2\ c2 == 0\)], "Output"] }, Open ]], Cell[TextData[{ "So ", Cell[BoxData[ \(TraditionalForm\`c\_1 = \ 1, \ c\_2\ = \ 1/2\)]], " and the final answer is:" }], "Text"], Cell[BoxData[ \(\(y[ t_]\ = \ \[ExponentialE]\^\(-t\)\ \((Cos[ 2 t]\ + \ \((1/2)\)\ Sin[2 t])\) + t\ \(\[ExponentialE]\^\(-t\)\) Sin[2 t];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[y[t], {t, 0, 2 \[Pi]}]\)], "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 %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.165539 0.43778 [ [.17539 .15304 -3 -9 ] [.17539 .15304 3 0 ] [.32696 .15304 -3 -9 ] [.32696 .15304 3 0 ] [.47854 .15304 -3 -9 ] [.47854 .15304 3 0 ] [.63011 .15304 -3 -9 ] [.63011 .15304 3 0 ] [.78169 .15304 -3 -9 ] [.78169 .15304 3 0 ] [.93327 .15304 -3 -9 ] [.93327 .15304 3 0 ] [.01131 .07798 -24 -4.5 ] [.01131 .07798 0 4.5 ] [.01131 .25309 -18 -4.5 ] [.01131 .25309 0 4.5 ] [.01131 .34065 -18 -4.5 ] [.01131 .34065 0 4.5 ] [.01131 .42821 -18 -4.5 ] [.01131 .42821 0 4.5 ] [.01131 .51576 -18 -4.5 ] [.01131 .51576 0 4.5 ] [.01131 .60332 -6 -4.5 ] [.01131 .60332 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 .16554 m .17539 .17179 L s [(1)] .17539 .15304 0 1 Mshowa .32696 .16554 m .32696 .17179 L s [(2)] .32696 .15304 0 1 Mshowa .47854 .16554 m .47854 .17179 L s [(3)] .47854 .15304 0 1 Mshowa .63011 .16554 m .63011 .17179 L s [(4)] .63011 .15304 0 1 Mshowa .78169 .16554 m .78169 .17179 L s [(5)] .78169 .15304 0 1 Mshowa .93327 .16554 m .93327 .17179 L s [(6)] .93327 .15304 0 1 Mshowa .125 Mabswid .05412 .16554 m .05412 .16929 L s .08444 .16554 m .08444 .16929 L s .11476 .16554 m .11476 .16929 L s .14507 .16554 m .14507 .16929 L s .2057 .16554 m .2057 .16929 L s .23602 .16554 m .23602 .16929 L s .26633 .16554 m .26633 .16929 L s .29665 .16554 m .29665 .16929 L s .35728 .16554 m .35728 .16929 L s .38759 .16554 m .38759 .16929 L s .41791 .16554 m .41791 .16929 L s .44822 .16554 m .44822 .16929 L s .50885 .16554 m .50885 .16929 L s .53917 .16554 m .53917 .16929 L s .56948 .16554 m .56948 .16929 L s .5998 .16554 m .5998 .16929 L s .66043 .16554 m .66043 .16929 L s .69074 .16554 m .69074 .16929 L s .72106 .16554 m .72106 .16929 L s .75138 .16554 m .75138 .16929 L s .81201 .16554 m .81201 .16929 L s .84232 .16554 m .84232 .16929 L s .87264 .16554 m .87264 .16929 L s .90295 .16554 m .90295 .16929 L s .96358 .16554 m .96358 .16929 L s .9939 .16554 m .9939 .16929 L s .25 Mabswid 0 .16554 m 1 .16554 L s .02381 .07798 m .03006 .07798 L s [(-0.2)] .01131 .07798 1 0 Mshowa .02381 .25309 m .03006 .25309 L s [(0.2)] .01131 .25309 1 0 Mshowa .02381 .34065 m .03006 .34065 L s [(0.4)] .01131 .34065 1 0 Mshowa .02381 .42821 m .03006 .42821 L s [(0.6)] .01131 .42821 1 0 Mshowa .02381 .51576 m .03006 .51576 L s [(0.8)] .01131 .51576 1 0 Mshowa .02381 .60332 m .03006 .60332 L s [(1)] .01131 .60332 1 0 Mshowa .125 Mabswid .02381 .09987 m .02756 .09987 L s .02381 .12176 m .02756 .12176 L s .02381 .14365 m .02756 .14365 L s .02381 .18743 m .02756 .18743 L s .02381 .20932 m .02756 .20932 L s .02381 .23121 m .02756 .23121 L s .02381 .27498 m .02756 .27498 L s .02381 .29687 m .02756 .29687 L s .02381 .31876 m .02756 .31876 L s .02381 .36254 m .02756 .36254 L s .02381 .38443 m .02756 .38443 L s .02381 .40632 m .02756 .40632 L s .02381 .4501 m .02756 .4501 L s .02381 .47198 m .02756 .47198 L s .02381 .49387 m .02756 .49387 L s .02381 .53765 m .02756 .53765 L s .02381 .55954 m .02756 .55954 L s .02381 .58143 m .02756 .58143 L s .02381 .05609 m .02756 .05609 L s .02381 .0342 m .02756 .0342 L s .02381 .01232 m .02756 .01232 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 .60332 m .02499 .60331 L .02605 .60327 L .02729 .6032 L .02846 .60311 L .03053 .60288 L .03279 .60252 L .03527 .602 L .0379 .60131 L .04262 .59966 L .04749 .59741 L .05205 .59476 L .06244 .58666 L .07305 .5753 L .08274 .56205 L .10458 .52211 L .12357 .4767 L .14429 .41778 L .18493 .28668 L .22406 .16485 L .24404 .11268 L .25455 .08919 L .26565 .06768 L .27559 .05147 L .28651 .03709 L .29644 .02717 L .30571 .02056 L .31072 .01805 L .31355 .01694 L .31616 .01612 L .31845 .01556 L .32096 .01511 L .32223 .01495 L .32289 .01488 L .32359 .01482 L .32488 .01475 L .32606 .01472 L .32723 .01472 L .32828 .01476 L .3295 .01483 L .33066 .01494 L .33172 .01507 L .3327 .01521 L .33493 .01563 L .33717 .01617 L .33964 .0169 L .34462 .0188 L .34983 .02137 L .35478 .02433 L .36407 .03113 L Mistroke .38496 .05135 L .42321 .09862 L .4639 .14941 L .48274 .1693 L .50309 .18682 L .51426 .19449 L .52454 .20026 L .53436 .20465 L .53978 .2066 L .54472 .20809 L .54927 .20924 L .55422 .21023 L .55694 .21067 L .55945 .211 L .562 .21128 L .5644 .21149 L .56644 .21162 L .56758 .21167 L .56863 .21171 L .56972 .21175 L .57091 .21177 L .57215 .21178 L .5733 .21177 L .57453 .21176 L .57569 .21173 L .57674 .2117 L .57787 .21165 L .58041 .2115 L .58274 .21132 L .58743 .21083 L .59246 .21011 L .60154 .20839 L .61234 .20569 L .6222 .20273 L .66201 .18786 L .70031 .17308 L .71967 .16667 L .74106 .1609 L .75079 .15878 L .76129 .15687 L .77113 .15542 L .78029 .15437 L .78985 .15357 L .7949 .15327 L .80031 .15303 L .80303 .15294 L .80447 .1529 L .80599 .15286 L .80732 .15284 L .80878 .15282 L Mistroke .80942 .15281 L .81011 .1528 L .81136 .15279 L .81264 .15279 L .81337 .15278 L .81405 .15278 L .81528 .15279 L .8166 .15279 L .8179 .15281 L .81912 .15282 L .8214 .15286 L .82381 .15291 L .82606 .15297 L .83117 .15315 L .83616 .15337 L .84149 .15366 L .85997 .15502 L .90135 .15925 L .94122 .16346 L .97619 .16636 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]HGAYHf4PAg9QL6QYHg`3oool00`000000oooo0?ooo`0U0?ooo`030000003o ool0oooo09`0oooo00070?ooo`040000003oool0oooo000000/0oooo00<000000?ooo`3oool00P3o ool00`000000oooo0?ooo`0k0?ooo`030000003oool0oooo02H0oooo00<000000?ooo`3oool0V`3o ool000L0oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`0000040?ooo`030000 003oool0oooo03X0oooo00<000000?ooo`3oool0:03oool00`000000oooo0?ooo`2J0?ooo`00203o ool2000000X0oooo0P0000050?ooo`030000003oool0oooo03X0oooo00<000000?ooo`3oool0:03o ool00`000000oooo0?ooo`2J0?ooo`006`3oool00`000000oooo0?ooo`0i0?ooo`030000003oool0 oooo02X0oooo00<000000?ooo`3oool0V@3oool001/0oooo0P00000j0?ooo`030000003oool0oooo 02/0oooo00<000000?ooo`3oool0V03oool001/0oooo00<000000?ooo`3oool0>03oool00`000000 oooo0?ooo`0]0?ooo`030000003oool0oooo09L0oooo000K0?ooo`030000003oool0oooo03P0oooo 00<000000?ooo`3oool0;P3oool00`000000oooo0?ooo`2F0?ooo`006`3oool00`000000oooo0?oo o`0g0?ooo`030000003oool0oooo0300oooo00<000000?ooo`3oool0U@3oool001/0oooo00<00000 0?ooo`3oool0=`3oool00`000000oooo0?ooo`0`0?ooo`030000003oool0oooo09D0oooo000K0?oo o`030000003oool0oooo03L0oooo00<000000?ooo`3oool0<@3oool00`000000oooo0?ooo`2D0?oo o`006`3oool2000003L0oooo00<000000?ooo`3oool0<`3oool00`000000oooo0?ooo`2C0?ooo`00 6`3oool00`000000oooo0?ooo`0S0?ooo`@000003`3oool00`000000oooo0?ooo`0B0?ooo`@00000 7P3oool00`000000oooo0?ooo`050?ooo`8000009`3oool00`000000oooo0?ooo`0S0?ooo`<00000 9`3oool200000180oooo000K0?ooo`030000003oool0oooo02D0oooo00<000000?ooo`3oool03@3o ool00`000000oooo0?ooo`0C0?ooo`030000003oool0oooo0200oooo00<000000?ooo`3oool00`3o ool010000000oooo0?ooo`00000V0?ooo`030000003oool0oooo02H0oooo00<000000?ooo`3oool0 8`3oool010000000oooo0?ooo`00000A0?ooo`006`3oool00`000000oooo0?ooo`0U0?ooo`030000 003oool0oooo00d0oooo00<000000?ooo`3oool0503oool00`000000oooo0?ooo`0P0?ooo`030000 003oool0oooo00D0oooo00<000000?ooo`3oool08@3oool5000002L0oooo00<000000?ooo`3oool0 8`3oool010000000oooo0?ooo`00000A0?ooo`006`3oool00`000000oooo0?ooo`0U0?ooo`030000 003oool0oooo00d0oooo00<000000?ooo`3oool05@3oool00`000000oooo0?ooo`0O0?ooo`030000 003oool0oooo00<0oooo0P00000T0?ooo`040000003oool0oooo000002D0oooo0`00000V0?ooo`<0 00004P3oool001/0oooo00<000000?ooo`3oool09@3oool00`000000oooo0?ooo`0<0?ooo`030000 003oool0oooo01L0oooo00<000000?ooo`3oool07`3oool00`000000oooo0?ooo`040?ooo`030000 003oool0oooo0280oooo00<000000?ooo`0000009@3oool00`000000oooo0?ooo`0V0?ooo`030000 003oool0oooo0180oooo000K0?ooo`800000903oool3000000h0oooo00<000000?ooo`3oool0503o ool010000000oooo0?ooo`00000R0?ooo`040000003oool0oooo00000080oooo00<000000?ooo`3o ool08`3oool2000002D0oooo00<000000?ooo`3oool09P3oool00`000000oooo0?ooo`0B0?ooo`00 6`3oool00`000000oooo0?ooo`0U0?ooo`030000003oool0oooo00`0oooo00<000000?ooo`3oool0 5@3oool2000002@0oooo00<000000?ooo`3oool00P00000W0?ooo`030000003oool0oooo02<0oooo 1000000V0?ooo`<000004@3oool001/0oooo00<000000?ooo`3oool0<`3oool00`000000oooo0?oo o`0m0?ooo`030000003oool0oooo0580oooo4@00000Y0?ooo`006`3oool00`000000oooo0?ooo`0c 0?ooo`030000003oool0oooo03h0oooo00<000000?ooo`3oool0B@3oool800000140oooo2000000Q 0?ooo`006`3oool00`000000oooo0?ooo`0b0?ooo`030000003oool0oooo0400oooo00<000000?oo o`3oool0@`3oool500000240oooo2`00000F0?ooo`006`3oool00`000000oooo0?ooo`0b0?ooo`03 0000003oool0oooo0400oooo00<000000?ooo`3oool0@03oool300000340oooo2P00000<0?ooo`00 503ooooo000000d00000000K0?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool01@3o ool00`000000oooo0?ooo`050?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool01@3o ool00`000000oooo0?ooo`050?ooo`050000003oool0oooo0?ooo`0000000`3oool00`000000oooo 0?ooo`050?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool01@3oool00`000000oooo 0?ooo`060?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool01@3oool00`000000oooo 0?ooo`050?ooo`030000003oool0oooo00D0oooo0P0000060?ooo`030000003oool0oooo00D0oooo 00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo00D0oooo 00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo0080oooo 0P0000000`3oool000000?ooo`060?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0 1@3oool00`000000oooo0?ooo`060?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0 1@3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0 1@3oool100000040oooo0@3oool001/0oooo00<000000?ooo`3oool0<@3oool00`000000oooo0?oo o`140?ooo`030000003oool0oooo03D0oooo0`00001?0?ooo`006`3oool00`000000oooo0?ooo`0a 0?ooo`030000003oool0oooo04D0oooo0P00000c0?ooo`800000DP3oool001/0oooo00<000000?oo o`3oool0<03oool00`000000oooo0?ooo`180?ooo`030000003oool0oooo02d0oooo0`00001D0?oo o`006`3oool200000340oooo00<000000?ooo`3oool0B@3oool00`000000oooo0?ooo`0Z0?ooo`80 0000E`3oool001/0oooo00<000000?ooo`3oool0<03oool00`000000oooo0?ooo`1:0?ooo`800000 9`3oool3000005T0oooo000K0?ooo`030000003oool0oooo0300oooo00<000000?ooo`3oool0C03o ool00`000000oooo0?ooo`0P0?ooo`@00000G03oool001/0oooo00<000000?ooo`3oool0;`3oool0 0`000000oooo0?ooo`1>0?ooo`800000703oool400000600oooo000K0?ooo`030000003oool0oooo 02l0oooo00<000000?ooo`3oool0D03oool2000001L0oooo0`00001T0?ooo`006`3oool00`000000 oooo0?ooo`0_0?ooo`030000003oool0oooo0580oooo0`00000A0?ooo`<00000I`3oool001/0oooo 0P00000_0?ooo`030000003oool0oooo05H0oooo100000090?ooo`@00000JP3oool001/0oooo00<0 00000?ooo`3oool0;P3oool00`000000oooo0?ooo`1J0?ooo`T00000KP3oool001/0oooo00<00000 0?ooo`3oool0;P3oool00`000000oooo0?ooo`3A0?ooo`006`3oool00`000000oooo0?ooo`0]0?oo o`030000003oool0oooo0=80oooo000K0?ooo`030000003oool0oooo02d0oooo00<000000?ooo`3o ool0dP3oool001/0oooo00<000000?ooo`3oool0;@3oool00`000000oooo0?ooo`3B0?ooo`006`3o ool2000002h0oooo00<000000?ooo`3oool0dP3oool001/0oooo00<000000?ooo`3oool0;03oool0 0`000000oooo0?ooo`3C0?ooo`006`3oool00`000000oooo0?ooo`0/0?ooo`030000003oool0oooo 0=<0oooo000K0?ooo`030000003oool0oooo02`0oooo00<000000?ooo`3oool0d`3oool000P0oooo 0P0000040?ooo`030000003oool0oooo0080oooo100000040?ooo`030000003oool0oooo02/0oooo 00<000000?ooo`3oool0e03oool000L0oooo00@000000?ooo`3oool00000203oool00`000000oooo 0?ooo`050?ooo`030000003oool0oooo02/0oooo00<000000?ooo`3oool0e03oool000L0oooo00@0 00000?ooo`3oool000002@3oool00`000000oooo0?ooo`040?ooo`800000;03oool00`000000oooo 0?ooo`3D0?ooo`001`3oool010000000oooo0?ooo`00000:0?ooo`030000003oool0oooo00<0oooo 00<000000?ooo`3oool0:P3oool00`000000oooo0?ooo`3E0?ooo`001`3oool010000000oooo0?oo o`00000;0?ooo`030000003oool0oooo0080oooo00<000000?ooo`3oool0:P3oool00`000000oooo 0?ooo`3E0?ooo`001`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo000000@0 oooo00<000000?ooo`3oool0:P3oool00`000000oooo0?ooo`3E0?ooo`00203oool2000000X0oooo 0P0000050?ooo`030000003oool0oooo02X0oooo00<000000?ooo`3oool0e@3oool001/0oooo00<0 00000?ooo`3oool0:@3oool00`000000oooo0?ooo`3F0?ooo`006`3oool2000002X0oooo00<00000 0?ooo`3oool0eP3oool001/0oooo00<000000?ooo`3oool0:@3oool00`000000oooo0?ooo`3F0?oo o`006`3oool00`000000oooo0?ooo`0X0?ooo`030000003oool0oooo0=L0oooo000K0?ooo`030000 003oool0oooo02P0oooo00<000000?ooo`3oool0e`3oool001/0oooo00<000000?ooo`3oool0:03o ool00`000000oooo0?ooo`3G0?ooo`006`3oool00`000000oooo0?ooo`0W0?ooo`030000003oool0 oooo0=P0oooo000K0?ooo`800000:03oool00`000000oooo0?ooo`3H0?ooo`006`3oool00`000000 oooo0?ooo`0W0?ooo`030000003oool0oooo0=P0oooo000K0?ooo`030000003oool0oooo02H0oooo 00<000000?ooo`3oool0f@3oool001/0oooo00<000000?ooo`3oool09P3oool00`000000oooo0?oo o`3I0?ooo`006`3oool00`000000oooo0?ooo`0V0?ooo`030000003oool0oooo0=T0oooo000K0?oo o`8000009P3oool00`000000oooo0?ooo`3J0?ooo`006`3oool00`000000oooo0?ooo`0U0?ooo`03 0000003oool0oooo0=X0oooo000K0?ooo`030000003oool0oooo02D0oooo00<000000?ooo`3oool0 fP3oool001/0oooo00<000000?ooo`3oool09@3oool00`000000oooo0?ooo`3J0?ooo`00203oool2 000000@0oooo00<000000?ooo`3oool0103oool00`000000oooo0?ooo`030?ooo`030000003oool0 oooo02@0oooo00<000000?ooo`3oool0f`3oool000L0oooo00@000000?ooo`3oool000002P3oool0 0`000000oooo0?ooo`030?ooo`030000003oool0oooo02@0oooo00<000000?ooo`3oool0f`3oool0 00L0oooo00@000000?ooo`3oool000001`3oool5000000@0oooo0P00000U0?ooo`030000003oool0 oooo0=/0oooo00070?ooo`040000003oool0oooo000000L0oooo00@000000?ooo`3oool000001@3o ool00`000000oooo0?ooo`0S0?ooo`030000003oool0oooo0=`0oooo00070?ooo`040000003oool0 oooo000000P0oooo00<000000?ooo`0000001@3oool00`000000oooo0?ooo`0S0?ooo`030000003o ool0oooo0=`0oooo00070?ooo`040000003oool0oooo000000T0oooo0P0000050?ooo`030000003o ool0oooo02<0oooo00<000000?ooo`3oool0g03oool000P0oooo0P00000;0?ooo`030000003oool0 oooo00<0oooo00<000000?ooo`3oool08P3oool00`000000oooo0?ooo`3M0?ooo`006`3oool00`00 0000oooo0?ooo`0R0?ooo`030000003oool0oooo0=d0oooo000K0?ooo`8000008`3oool00`000000 oooo0?ooo`3M0?ooo`006`3oool00`000000oooo0?ooo`0Q0?ooo`030000003oool0oooo0=h0oooo 000K0?ooo`030000003oool0oooo0240oooo00<000000?ooo`3oool0gP3oool001/0oooo00<00000 0?ooo`3oool08@3oool00`000000oooo0?ooo`3N0?ooo`006`3oool00`000000oooo0?ooo`0P0?oo o`030000003oool0oooo0=l0oooo000K0?ooo`030000003oool0oooo0200oooo00<000000?ooo`3o ool0g`3oool001/0oooo0P00000Q0?ooo`030000003oool0oooo0=l0oooo000K0?ooo`030000003o ool0oooo0200oooo00<000000?ooo`3oool0g`3oool001/0oooo00<000000?ooo`3oool07`3oool0 0`000000oooo0?ooo`3P0?ooo`006`3oool00`000000oooo0?ooo`0O0?ooo`030000003oool0oooo 0>00oooo000K0?ooo`030000003oool0oooo01l0oooo00<000000?ooo`3oool0h03oool001/0oooo 00<000000?ooo`3oool07P3oool00`000000oooo0?ooo`3Q0?ooo`006`3oool2000001l0oooo00<0 00000?ooo`3oool0h@3oool001/0oooo00<000000?ooo`3oool07P3oool00`000000oooo0?ooo`3Q 0?ooo`006`3oool00`000000oooo0?ooo`0M0?ooo`030000003oool0oooo0>80oooo00080?ooo`80 0000103oool00`000000oooo0?ooo`030?ooo`8000001@3oool00`000000oooo0?ooo`0M0?ooo`03 0000003oool0oooo0>80oooo00070?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3o ool00000103oool00`000000oooo0?ooo`0M0?ooo`030000003oool0oooo0>80oooo00070?ooo`04 0000003oool0oooo000000P0oooo00@000000?ooo`3oool00000103oool2000001d0oooo00<00000 0?ooo`3oool0h`3oool000L0oooo00@000000?ooo`3oool00000203oool3000000D0oooo00<00000 0?ooo`3oool0703oool00`000000oooo0?ooo`3S0?ooo`001`3oool010000000oooo0?ooo`000008 0?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool06`3oool00`000000oooo0?ooo`3T 0?ooo`001`3oool010000000oooo0?ooo`0000080?ooo`030000003oool0oooo00D0oooo00<00000 0?ooo`3oool06`3oool00`000000oooo0?ooo`3T0?ooo`00203oool2000000X0oooo0`0000040?oo o`030000003oool0oooo01/0oooo00<000000?ooo`3oool0i03oool001/0oooo00<000000?ooo`3o ool06P3oool00`000000oooo0?ooo`3U0?ooo`006`3oool2000001/0oooo00<000000?ooo`3oool0 i@3oool001/0oooo00<000000?ooo`3oool06@3oool00`000000oooo0?ooo`3V0?ooo`006`3oool0 0`000000oooo0?ooo`0I0?ooo`030000003oool0oooo0>H0oooo000K0?ooo`030000003oool0oooo 01T0oooo00<000000?ooo`3oool0iP3oool001/0oooo00<000000?ooo`3oool0603oool00`000000 oooo0?ooo`3W0?ooo`006`3oool00`000000oooo0?ooo`0H0?ooo`030000003oool0oooo0>L0oooo 000K0?ooo`800000603oool00`000000oooo0?ooo`3X0?ooo`006`3oool00`000000oooo0?ooo`0G 0?ooo`030000003oool0oooo0>P0oooo000K0?ooo`030000003oool0oooo01L0oooo00<000000?oo o`3oool0j03oool001/0oooo00<000000?ooo`3oool05P3oool00`000000oooo0?ooo`3Y0?ooo`00 6`3oool00`000000oooo0?ooo`0F0?ooo`030000003oool0oooo0>T0oooo000K0?ooo`030000003o ool0oooo01D0oooo00<000000?ooo`3oool0jP3oool001/0oooo0P00000F0?ooo`030000003oool0 oooo0>X0oooo000K0?ooo`030000003oool0oooo01D0oooo00<000000?ooo`3oool0jP3oool001/0 oooo00<000000?ooo`3oool0503oool00`000000oooo0?ooo`3[0?ooo`006`3oool00`000000oooo 0?ooo`0D0?ooo`030000003oool0oooo0>/0oooo00080?ooo`800000103oool00`000000oooo0?oo o`030?ooo`8000001@3oool00`000000oooo0?ooo`0C0?ooo`030000003oool0oooo0>`0oooo0007 0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool00000103oool00`000000oooo 0?ooo`0C0?ooo`030000003oool0oooo0>`0oooo00070?ooo`040000003oool0oooo000000P0oooo 00@000000?ooo`3oool00000103oool2000001<0oooo00<000000?ooo`3oool0k@3oool000L0oooo 00@000000?ooo`3oool000002@3oool2000000D0oooo00<000000?ooo`3oool04P3oool00`000000 oooo0?ooo`3]0?ooo`001`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo0000 00@0oooo00<000000?ooo`3oool04P3oool00`000000oooo0?ooo`3]0?ooo`001`3oool010000000 oooo0?ooo`0000080?ooo`040000003oool0oooo000000@0oooo00<000000?ooo`3oool04@3oool0 0`000000oooo0?ooo`3^0?ooo`00203oool2000000X0oooo0P0000050?ooo`030000003oool0oooo 0140oooo00<000000?ooo`3oool0kP3oool001/0oooo00<000000?ooo`3oool0403oool00`000000 oooo0?ooo`3_0?ooo`006`3oool200000140oooo00<000000?ooo`3oool0k`3oool001/0oooo00<0 00000?ooo`3oool03`3oool00`000000oooo0?ooo`3`0?ooo`006`3oool00`000000oooo0?ooo`0? 0?ooo`030000003oool0oooo0?00oooo000K0?ooo`030000003oool0oooo00h0oooo00<000000?oo o`3oool0l@3oool001/0oooo00<000000?ooo`3oool03P3oool00`000000oooo0?ooo`3a0?ooo`00 6`3oool2000000h0oooo00<000000?ooo`3oool0lP3oool001/0oooo00<000000?ooo`3oool03@3o ool00`000000oooo0?ooo`3b0?ooo`006`3oool00`000000oooo0?ooo`0<0?ooo`030000003oool0 oooo0?<0oooo000K0?ooo`030000003oool0oooo00`0oooo00<000000?ooo`3oool0l`3oool001/0 oooo00<000000?ooo`3oool02`3oool00`000000oooo0?ooo`3d0?ooo`006`3oool00`000000oooo 0?ooo`0:0?ooo`030000003oool0oooo0?D0oooo000K0?ooo`8000002P3oool00`000000oooo0?oo o`3f0?ooo`006`3oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo0?L0oooo000K0?oo o`030000003oool0oooo00L0oooo00<000000?ooo`3oool0n03oool001/0oooo00<000000?ooo`3o ool01@3oool200000?/0oooo000C0?ooo`@00000103oool00`000000oooo0?ooo`030?ooo`800000 o@3oool001D0oooo00<000000?ooo`3oool00`3oool010000000oooo0?ooo`3oool200000?l0oooo 000E0?ooo`030000003oool0oooo00<0oooo1000003o0?ooo`80oooo000E0?ooo`030000003oool0 oooo00<0oooo00<000000?ooo`3oool0o`3oool30?ooo`005@3oool00`000000oooo0?ooo`030?oo o`030000003oool0oooo0?l0oooo0`3oool001<0oooo0`0000050?ooo`030000003oool0oooo0?l0 oooo0`3oool001D0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`3o0?ooo`<0oooo 003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40oooo003o0?ooob40oooo 003o0?ooob40oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -0.429445, \ 0.0248015, 0.00858722}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{520, 600}, WindowMargins->{{32, Automatic}, {Automatic, 8}} ] (******************************************************************* 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[CellGroupData[{ Cell[1727, 52, 39, 0, 45, "Subsection"], Cell[1769, 54, 300, 12, 32, "Text"], Cell[2072, 68, 67, 1, 32, "Text"], Cell[2142, 71, 42, 0, 32, "Text"], Cell[CellGroupData[{ Cell[2209, 75, 77, 1, 27, "Input"], Cell[2289, 78, 125, 2, 27, "Output"] }, Open ]], Cell[2429, 83, 141, 3, 28, "Input"], Cell[CellGroupData[{ Cell[2595, 90, 55, 1, 27, "Input"], Cell[2653, 93, 153, 2, 28, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2843, 100, 56, 1, 27, "Input"], Cell[2902, 103, 240, 4, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3179, 112, 91, 1, 27, "Input"], Cell[3273, 115, 106, 2, 28, "Output"] }, Open ]], Cell[3394, 120, 443, 15, 50, "Text"], Cell[3840, 137, 196, 4, 28, "Input"], Cell[4039, 143, 35, 0, 32, "Text"], Cell[CellGroupData[{ Cell[4099, 147, 48, 1, 27, "Input"], Cell[4150, 150, 41, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4228, 156, 53, 1, 27, "Input"], Cell[4284, 159, 54, 1, 27, "Output"] }, Open ]], Cell[4353, 163, 141, 5, 32, "Text"], Cell[4497, 170, 201, 4, 28, "Input"], Cell[CellGroupData[{ Cell[4723, 178, 61, 1, 27, "Input"], Cell[4787, 181, 21249, 584, 186, 5594, 386, "GraphicsData", "PostScript", \ "Graphics"], Cell[26039, 767, 130, 3, 27, "Output"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)