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Check that your solutions are correct by comparing ", StyleBox["X'[t]", "Input"], " and ", StyleBox["A.X[t]", "Input"], ". Superimpose several plots for each using ", StyleBox["ParametricPlot3D[]", "Input"], " or ", StyleBox["ParametricPlot3D[]", "Input"], ".\n\n", StyleBox["a", FontSlant->"Italic"], ") ", StyleBox["A = {{ 3, 2},\n {-1, 1}};", "Input"], "\n\n", StyleBox["b", FontSlant->"Italic"], ") ", StyleBox["A = {{1,-1, 4},\n {3, 2,-1},\n {2, 1,-1}};", "Input"], "\n \n", StyleBox["c", FontSlant->"Italic"], ") ", StyleBox[ "A = {{-14,18, 8},\n { -6, 8, 2},\n { -5, 7, 3}};", "Input"] }], "Text"], Cell["Solution", "Subsubsection"], Cell["", "Input"], Cell[CellGroupData[{ Cell["Problem 2", "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"], ". Verify that ", StyleBox["Phi'[t]==A.Phi[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 \n", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"] }, Open ]], Cell["Solution", "Subsubsection"], Cell["", "Input"], Cell[CellGroupData[{ Cell["Problem 3", "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. Check that your solution is correct by comparing ", StyleBox["X'[t]", "Input"], " and ", StyleBox["A.X[t]+G[t]", "Input"], ". Then find the solution that satisfies the given initial condition.\n\n", StyleBox["a)", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Courier", FontWeight->"Bold"], "(see ", StyleBox["Problem 2", 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"], "\n\n", StyleBox["b) ", FontSlant->"Italic"], "(see ", StyleBox["Problem 2", FontWeight->"Bold"], StyleBox["b", FontSlant->"Italic"], ")", StyleBox["\n", 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[" G[t_] = {1,Cos[t],Sin[t]};\n X[0]=={1,1,-2};\n \n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["c)", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Courier", FontWeight->"Bold"], "(see ", StyleBox["Problem 2", FontWeight->"Bold"], StyleBox["c", FontSlant->"Italic"], ")\n", StyleBox[" ", "Input"], 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 G[t_] = {E^t,1,E^(-t)};\n X[0]=={1,1,1};", FontFamily->"Courier", FontWeight->"Bold"] }], "Text"] }, Open ]], Cell["Solution", "Subsubsection"], Cell["", "Input"], Cell[CellGroupData[{ Cell["Problem 4", "Subsubsection"], Cell[TextData[{ "Consider the autonomous system:\n", StyleBox[" x'[t] == 1 - x[t]y[t]\n y'[t] == x[t] - y[t]^3\n", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["a)", FontSlant->"Italic"], " Find all critical points of this system. At each critical point, \ calculate the corresponding linear system and find the eigenvalues of the \ coefficient matrix. Then identify the type and stability of the critical \ point.\n", StyleBox["b)", FontSlant->"Italic"], " Plot the vector field using ", StyleBox["PlotVectorField[]", FontFamily->"Courier", FontWeight->"Bold"], " with the options ", StyleBox["ScaleFunction->(Log[50#+1]&)", FontFamily->"Courier", FontWeight->"Bold"], " (this makes the arrows bigger near the critical points) and ", StyleBox["Frame->True", FontFamily->"Courier", FontWeight->"Bold"], ". Use the vector field plot to choose a representative sample of several \ initial points ", StyleBox["{x0,y0}", FontFamily->"Courier", FontWeight->"Bold"], ". Use ", StyleBox["NDSolve[]", FontFamily->"Courier", FontWeight->"Bold"], " to obtain numerical solutions through these points and plot the \ corresponding trajectories on a single graph. Indicate what happens to the \ trajectories for increasing ", StyleBox["t", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text"] }, Open ]], Cell["Solution", "Subsubsection"], Cell["", "Input"], Cell["Problem 5", "Subsubsection"], Cell[TextData[{ "In this problem we consider the equations describing the motion of a \ pendulum, both damped and undamped. For each situation below convert the \ equation into a system of equations in ", StyleBox["x[t] = theta[t]", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["y[t] = theta'[t]", FontFamily->"Courier", FontWeight->"Bold"], ". Plot the corresponding vector field. Use ", StyleBox["NDSolve[]", FontFamily->"Courier", FontWeight->"Bold"], " to solve this system and plot on a single graph the trajectories \ corresponding to the given initial velocities ", StyleBox["b", FontFamily->"Courier", FontWeight->"Bold"], ". Based on these plots, describe what the pendulum is doing in each case. \ Experiment to see if you can find a value for the initial velocity ", StyleBox["b", FontFamily->"Courier", FontWeight->"Bold"], " such that the trajectory approaches the unstable equilibrium point ", StyleBox["{Pi,0}", FontFamily->"Courier", FontWeight->"Bold"], " corresponding to the pendulum being stationary in an upright position.\n\n\ ", StyleBox["a)", FontSlant->"Italic"], " The undamped pendulum:\n", StyleBox[ "\n theta''[t] + Sin[theta] == 0,\n theta[0] == 0, theta'[0] == b\n\n\ ", FontFamily->"Courier", FontWeight->"Bold"], "The initial velocities are ", StyleBox["b = 0.5", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["3.0", FontFamily->"Courier", FontWeight->"Bold"], " in steps of ", StyleBox["0.5", FontFamily->"Courier", FontWeight->"Bold"], ".\n", StyleBox["\nb)", FontSlant->"Italic"], " The damped pendulum:\n", StyleBox[ "\n theta''[t] + 0.5 theta'[t] + Sin[theta] == 0,\n theta[0] == 0, \ theta'[0] == b\n\n", FontFamily->"Courier", FontWeight->"Bold"], "The initial velocities are ", StyleBox["b = 0.5", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["6.0", FontFamily->"Courier", FontWeight->"Bold"], " in steps of ", StyleBox["0.5", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["Solution", "Subsubsection"], Cell["", "Input"] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1024}, {0, 748}}, AutoGeneratedPackage->None, CellGrouping->Manual, WindowSize->{653, 646}, WindowMargins->{{Automatic, 7}, {Automatic, 1}}, MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@0000000000000006P801T1T00000000000 00000000000000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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