(*********************************************************************** 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). 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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[ 8656, 335]*) (*NotebookOutlinePosition[ 9334, 360]*) (* CellTagsIndexPosition[ 9290, 356]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 325: Differential Equations\tAssignment \tFall 1998", "Subsection"], Cell["Name:", "Subsection"], Cell["Section: ", "Subsection"], Cell["Professor: ", "Subsection"], Cell["\<\ This assignment is due Monday, Nov 23, 1998. Print out your \ answer to the problems. This assignment will be counted as one homework.\ \>", "Subsection"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection"], Cell["<True] }, Open ]], Cell[CellGroupData[{ Cell["Problem 1", "Subsubsection"], Cell[TextData[{ "Find the indefinite integral of the following functions\n", StyleBox["\tE^t Cos[t]^3 Sin[t]^5; (t-1)^10 Log[t+1]", "Input"] }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "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["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 3", "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"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "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["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "Solve the Differential Equation \n", StyleBox[" y'''[t] + 17y''[t] + 22y'[t] -40y[t] == 0\n", "Input"], "with the initial conditions:\n", StyleBox["\ty[0] == 1, y'[0] == 99, y''[0]==999, \n", "Input"], "using ", StyleBox["DSolve[]", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 6", "Subsubsection"], Cell[TextData[{ "Find the solutions to the following differential equation:\n", StyleBox[ "\ty''''[t] + 2y'''[t] + 5y''[t] + 8y'[t] + 4y[t]==Sin[2t] + t*E^(-t)\n", "Input"], "with the initial conditions\n", StyleBox["\ty[0] == y0, y'[0] == 0, y''[0] == 0, y'''[0] == 0\n", "Input"], "Superimpose plots of the particular solutions", StyleBox[" ", "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["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 7", "Subsubsection"], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Find the Laplace transforms of the functions ", StyleBox["t*E^(3t)", "Input"], " and ", StyleBox["t^2 Sin[5t]", "Input"], " by using ", StyleBox["LaplaceTransform[]", "Input"], ".\n", StyleBox["b) ", FontSlant->"Italic"], "Find the inverse Laplace transform of the functions ", StyleBox["(s-1)/(s^2-4)", "Input"], " and\n", StyleBox["(s^2 + s + 5)/(s^3 - 4s^2 + 5s)", "Input"], " by using ", StyleBox["InverseLaplaceTransform[]", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 8", "Subsubsection"], Cell[TextData[{ "Solve the initial value problems using the Laplace transform. Compare the \ answers with the results given by ", StyleBox["DSolve[]", "Input"], " and plot the solutions:\n ", StyleBox["y''[t] + 9y[t] == Cos[2t], y[0] == 1, y'[0] == 0", "Input"], "\n" }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 9", "Subsubsection"], Cell[TextData[{ "For each of the matrices ", StyleBox["A", "Input"], " below, find the fundamental matrix ", StyleBox["Phi[t]", "Input"], " such that ", StyleBox["Phi[0]", "Input"], " is the identity matrix for the system ", StyleBox["X'[t] == A.X[t]", "Input"], ". Then use ", StyleBox["Phi[t]", "Input"], " to find a solution satisfying the given initial condition.\n\n", StyleBox["a)", FontSlant->"Italic"], StyleBox[" A =", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", "Input", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "{{-7,-8, 2},\n { 9,11,-3},\n { 9, 8, 0}}", "Input"], StyleBox[";\n X[0]=={3,-1,1};", FontFamily->"Courier", FontWeight->"Bold"], "\n\n", StyleBox["b)", FontSlant->"Italic"], StyleBox[" ", "Input", FontSlant->"Italic"], StyleBox["A = ", "Input", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "{{-4,-2,-1},\n { 2, 6,-3},\n { 2, 4,-1}}", "Input"], StyleBox[";\n", "Input", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" X[0]=={1,1,-2};\n \n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["c)", FontSlant->"Italic"], StyleBox[" A =", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", "Input", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "{{ 5, 3,-1},\n {-16,-6,-1},\n { -4,-3, 2}}", "Input"], StyleBox[";\n X[0]=={1,1,1}; \n", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"], Cell["Solution", "Subsubsection"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 10", "Subsubsection"], Cell[TextData[{ "Use the method of variation of parameters to find the general solution of \ the system ", StyleBox["X'[t] == A.X[t] + G[t]", "Input"], " for each of the matrices ", StyleBox["A", "Input"], " and vector functions ", StyleBox["G[t]", "Input"], " below. . Then find the solution that satisfies the given initial \ condition.\n\n(see ", StyleBox["Problem 9", FontWeight->"Bold"], StyleBox["a", FontSlant->"Italic"], ")", StyleBox["\n A =", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" ", "Input", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ "{{-7,-8, 2},\n { 9,11,-3},\n { 9, 8, 0}}", "Input"], StyleBox[";\n G[t_] = {t^2,t,1};\n X[0]=={3,-1,1};", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"], Cell["Solution", "Subsubsection"] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1152}, {0, 900}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{747, 640}, WindowMargins->{{133, Automatic}, {Automatic, 51}} ] (*********************************************************************** 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. 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