(*********************************************************************** 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[ 10076, 396]*) (*NotebookOutlinePosition[ 10986, 426]*) (* CellTagsIndexPosition[ 10942, 422]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 325: Differential Equations\tAssignment 2\t\tFall 1997", "Subsection"], Cell["\<\ Names (up to 3 students): \ \>", "Subsection"], Cell["Section:", "Subsection"], Cell[CellGroupData[{ Cell["Initialization", "Subsubsection"], Cell["<True] }, Open ]], Cell[CellGroupData[{ Cell["Problem 1", "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["Integrate[]", "Input"], ". Compare the result with ", 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 partial fraction decompositions (use ", StyleBox["Apart[]", "Input"], ") and the tables on p.300 of the textbook. Compare the result with ", StyleBox["InverseLaplaceTransform[]", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2", "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["a)", FontSlant->"Italic"], " ", StyleBox["y''[t] + 9y[t] == Cos[2t], y[0] == 1, y'[0] == 0", "Input"], "\n", StyleBox["b)", FontSlant->"Italic"], " ", StyleBox[ "y''''[t] - y[t] ==0, y[0] ==-1, y'[0] == 0, y''[0] == 0, y'''[0] == 1", "Input"], "\n" }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 3", "Subsubsection"], Cell[TextData[{ "Solve the initial value problems using the Laplace transform and plot the \ solutions:\n", StyleBox["a)", FontSlant->"Italic"], StyleBox[ " y''''[t] + 2y'''[t] + 2y''[t] + 2y'[t] + y[t] == 10 UnitStep[t-5],\n\t\ y[0] == 0, y'[0] == 0, y''[0]==0, y'''[0]==0\n", "Input"], StyleBox["b)", FontSlant->"Italic"], StyleBox[ " y''''[t] + 8y''[t] + 16y[t] == DiracDelta[t-Pi] - DiracDelta[t-4Pi],\n\t\ y[0] == 0, y'[0] == 0, y''[0]==0, y'''[0]==0", "Input"] }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "Consider the initial value problem\n", StyleBox[ "\ty''[t] + 1/3 y'[t] + 4y[t] == f[k,t], y[0] == 0, y'[0] == 0\n", "Input"], "where ", StyleBox["f[k_,t_] = 1/(2k)", "Input"], " if ", StyleBox["4-k <= t < 4+k", "Input"], " and is ", StyleBox["0", "Input"], " otherwise.", StyleBox["\n", "Input"], "Solve the equation using Laplace transforms by writing ", StyleBox["f[k,t]", "Input"], " in terms of the unit step function.\nSuperimpose plots of ", StyleBox["f[k,t]", "Input"], " for ", StyleBox["k == 3/2", "Input"], ", ", StyleBox["1", "Input"], ", and ", StyleBox["1/2", "Input"], " and then superimpose plots of the solutions to the initial value problem \ for the same values of ", StyleBox["k", "Input"], ". Try to determine the limiting the solution as ", StyleBox["k", "Input"], " approaches ", StyleBox["0", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"] }, Open ]], Cell["", "Input"], Cell[CellGroupData[{ Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "Consider the initial value problem\n", StyleBox[ "\ty''[t] + h y'[t] + y[t] == k*DiracDelta[t-1],\n\ty[0] == 0, y'[0] == 0\n\ ", "Input"], "where ", StyleBox["k", "Input"], " is the magnitude of an impulse at ", StyleBox["t==1", "Input"], " and ", StyleBox["h", "Input"], " is the damping coefficient (or resistance).\n", StyleBox["a)", FontSlant->"Italic"], " Let ", StyleBox["h==0.5", "Input"], ". Find the value of ", StyleBox["k", "Input"], " for which the response has a peak value of ", StyleBox["2", "Input"], ". Call this value ", StyleBox["k0", "Input"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Repeat part ", StyleBox["a)", FontSlant->"Italic"], " for ", StyleBox["h==0.25", "Input"], ".\n", StyleBox["c)", FontSlant->"Italic"], " Determine how ", StyleBox["k0", "Input"], " varies as ", StyleBox["h", "Input"], " decreases. What is the value of ", StyleBox["k0", "Input"], " when ", StyleBox["h==0", "Input"], "?" }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 6", "Subsubsection"], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Use the Convolution Theorem 6.6.1 (p.326) to express the inverse Laplace \ transform of the function\n", StyleBox["H[s_] = s/((s+1)^2(s^2+9))", "Input"], " in terms of an integral. Evaluate this integral with ", StyleBox["Integrate[]", "Input"], " and compare the final result with ", StyleBox["InverseLaplaceTransform[H[s],s,t]", "Input"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Express the solution of the initial value problem\n", StyleBox[ "\ty''''[t] + 10y''[t] + 9y[t] == g[t],\n\ty[0]==2, y'[0]==1, y''[0]==0, \ y'''[0]==0\n", "Input"], "in terms of a convolution integral. (Use ", StyleBox["LaplaceTransform[]", "Input"], " and ", StyleBox["InverseLaplaceTransform[]", "Input"], ").\n", StyleBox["c)", FontSlant->"Italic"], " Use the result of ", StyleBox["b)", FontSlant->"Italic"], " to generate solutions to the initial value problem when ", StyleBox["g[t_]", "Input"], " is ", StyleBox["Cos[t]", "Input"], ", ", StyleBox["E^t", "Input"], ", and ", StyleBox["t^2", "Input"], ". (You should only need to make a substitution and carry out the \ integration)." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 7", "Subsubsection"], Cell[TextData[{ StyleBox["a)", FontSlant->"Italic"], " Find the eigenvalues of the matrix\n", StyleBox[ "\tA = {{ 7,-3, 3},\n\t {-1, 5, 1},\n\t { 2,-2, 8}}\n", "Input"], "by using ", StyleBox["Det[]", "Input"], ". Then find the eigenvectors corresponding to each eigenvalue ", StyleBox["r", "Input"], " by solving the appropriate linear system.\nCompare your results to the \ output of ", StyleBox["Eigensystem[]", "Input"], ".\n", StyleBox["b)", FontSlant->"Italic"], " Find a matrix ", StyleBox["T", "Input"], " such that ", StyleBox["Inverse[T].A.T == d", "Input"], " where ", StyleBox["d", "Input"], " is a diagonal matrix (see p. 362). Verify your choice of ", StyleBox["T", "Input"], " by calculating ", StyleBox["Inverse[T].A.T", "Input"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 8", "Subsubsection"], Cell[TextData[{ "Consider the vector functions\n", StyleBox[ "\tX1[t_] = { t , t^2, t^3};\n\tX2[t_] = { t , t^2, 1 };\n\tX3[t_] = {t^2, \ 1 , t };\n", "Input"], StyleBox["a)", FontSlant->"Italic"], " For what values of ", StyleBox["t", "Input"], " are the ", StyleBox["vectors", FontSlant->"Italic"], " ", StyleBox["X1[t]", "Input"], ", ", StyleBox["X2[t]", "Input"], ", ", StyleBox["X3[t]", "Input"], " linearly independent?\n", StyleBox["b)", FontSlant->"Italic"], " On what intervals are the ", StyleBox["vector functions ", FontSlant->"Italic"], StyleBox["X1[t]", "Input"], ", ", StyleBox["X2[t]", "Input"], ", ", StyleBox["X3[t]", "Input"], " linearly independent", StyleBox["?", FontSlant->"Italic"], "\n", StyleBox["c)", FontSlant->"Italic"], " What conclusion can be draw about the coefficients in a system of \ homogeneous differential equations satisfied by\n ", StyleBox["X1", "Input"], ", ", StyleBox["X2", "Input"], ", ", StyleBox["X3", "Input"], "?\n", StyleBox["d)", FontSlant->"Italic"], " Find such a system and verify the conclusions of part ", StyleBox["c)", FontSlant->"Italic"], "." }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, Evaluator->"Local", CellGrouping->Manual, WindowSize->{639, 658}, WindowMargins->{{52, Automatic}, {Automatic, 4}}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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