(*********************************************************************** 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[ 310459, 10336]*) (*NotebookOutlinePosition[ 311447, 10369]*) (* CellTagsIndexPosition[ 311403, 10365]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Math 325: Differential Equations\tAssignment 2 Solutions\t\tFall \ 1997\ \>", "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[TextData[StyleBox["a)", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Integrate[t*E^(3t)E^(-s*t), {t,0,Infinity}]", "Text"]], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{\(Re[s] > 3\), ",", \(1\/\((\(-3\) + s)\)\^2\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(E\^\(3\ t - s\ t\)\ t\) \[DifferentialD]t\)}]}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "LaplaceTransform[", StyleBox["t*E^(3t)", "Text"], ",t,s]" }], "Input"], Cell[BoxData[ \(1\/\((\(-3\) + s)\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Integrate[", "Text"], StyleBox["t^2 Sin[5t]", "Input"], StyleBox["E^(-s*t), {t,0,Infinity}]", "Text"] }], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ \(Re[s] > 0\), ",", \(\(10\ \((\(-25\) + 3\ s\^2)\)\)\/\((25 + s\^2)\)\^3\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", "0", InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(E\^\(\(-s\)\ t\)\ t\^2\ Sin[5\ t]\) \[DifferentialD]t\)}]}], "]"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "LaplaceTransform[", StyleBox["t^2 Sin[5t]", "Input"], ",t,s]" }], "Input"], Cell[BoxData[ \(\(40\ s\^2\)\/\((25 + s\^2)\)\^3 - 10\/\((25 + s\^2)\)\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Together[%]", "Input"], Cell[BoxData[ \(\(10\ \((\(-25\) + 3\ s\^2)\)\)\/\((25 + s\^2)\)\^3\)], "Output"] }, Open ]], Cell[TextData[StyleBox["b)", FontSlant->"Italic"]], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "Apart[", StyleBox["(s-1)/(s^2-4)", "Input"], "]" }], "Input"], Cell[BoxData[ \(1\/\(4\ \((\(-2\) + s)\)\) + 3\/\(4\ \((2 + s)\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "By the table on p. 300, the inverse Laplace transform is\n", StyleBox[ "(1/4)E^(2t) + (3/4)E^(-2t)\n\t== (1/2)(E^(2t) + E^(-2t)) - (1/4)(E^(2t) - \ E^(-2t))\n\t== Cosh[2t] - (1/2)Sinh[2t]", "Input"] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "InverseLaplaceTransform[", StyleBox["(s-1)/(s^2-4),s,t", "Input"], "]" }], "Input"], Cell[BoxData[ \(Cosh[2\ t] - 1\/2\ Sinh[2\ t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "Apart[(s^2 + s + 5)/(s^3 - 4s^2 + 5s)]", "Input"]], "Input"], Cell[BoxData[ \(1\/s + 5\/\(5 - 4\ s + s\^2\)\)], "Output"] }, Open ]], Cell[TextData[{ "Since ", StyleBox["s^2 - 4s + 5 == (s-2)^2 + 1", "Input"], ", the tables give the inverse Laplace transform as\n", StyleBox["1 + 5 E^(2t)Sin[t]", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ "InverseLaplaceTransform[", StyleBox["(s^2 + s + 5)/(s^3 - 4s^2 + 5s),s,t]", "Input"] }], "Input"], Cell[BoxData[ \(1 + 5\ E\^\(2\ t\)\ Sin[t]\)], "Output"] }, Open ]] }, 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[TextData[StyleBox["a)", FontSlant->"Italic"]], "Text"], Cell["Y[s_] := LaplaceTransform[y[t],t,s]", "Input"], Cell[CellGroupData[{ Cell[TextData[{ "eqn = LaplaceTransform[", StyleBox["y''[t] + 9y[t] == Cos[2t]", "Input"], ",t,s]" }], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ \(9\ LaplaceTransform[y[t], t, s]\), "+", \(s\^2\ LaplaceTransform[y[t], t, s]\), "-", \(s\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "==", \(s\/\(4 + s\^2\)\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tran = Solve[eqn/.{y[0]->1, 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"\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[TextData[StyleBox["a)", FontSlant->"Italic"]], "Text"], Cell["Y[s_] := LaplaceTransform[y[t],t,s]", "Input"], Cell[CellGroupData[{ Cell[TextData[{ "eqn = LaplaceTransform[\n\t", StyleBox["y''[t] + 0.5 y'[t] + y[t] == k*DiracDelta[t-1]", "Input"], ",t,s]" }], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ \(LaplaceTransform[y[t], t, s]\), "+", \(s\^2\ LaplaceTransform[y[t], t, s]\), "+", RowBox[{ StyleBox["0.5`", StyleBoxAutoDelete->True, PrintPrecision->1], " ", \((s\ LaplaceTransform[y[t], t, s] - y[0])\)}], "-", \(s\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "==", \(E\^\(-s\)\ k\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tran = Solve[eqn/.{y[0]->0, y'[0]->0},Y[s]]", "Input"], Cell[BoxData[ \({{LaplaceTransform[y[t], t, s] \[Rule] \(1.`\ E\^\(-s\)\ k\)\/\(\(1.`\[InvisibleSpace]\) + 0.5`\ s + s\^2\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["soln = Chop[InverseLaplaceTransform[Y[s]/.First[tran],s,t]]", "Input"], Cell[BoxData[ \(1.03279555898864439`\ E\^\(\(-0.25`\)\ \((\(-1\) + t)\)\)\ k\ Sin[0.96824583655185421`\ \((\(-1\) + t)\)]\ UnitStep[\(-1\) + t]\)], "Output"] }, Open ]], Cell[TextData[{ "Let's look at the graph for ", StyleBox["k==1", "Input"], ". 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Cell[CellGroupData[{ Cell[TextData[{ "eqn = LaplaceTransform[\n\t", StyleBox["y''[t] + y[t] == k*DiracDelta[t-1]", "Input"], ",t,s]" }], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ \(LaplaceTransform[y[t], t, s]\), "+", \(s\^2\ LaplaceTransform[y[t], t, s]\), "-", \(s\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "==", \(E\^\(-s\)\ k\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["tran = Solve[eqn/.{y[0]->0, y'[0]->0},Y[s]]", "Input"], Cell[BoxData[ \({{LaplaceTransform[y[t], t, s] \[Rule] \(E\^\(-s\)\ k\)\/\(1 + s\^2\)}} \)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["soln = InverseLaplaceTransform[Y[s]/.First[tran],s,t]", "Input"], Cell[BoxData[ \(\(-k\)\ Sin[1 - t]\ UnitStep[\(-1\) + t]\)], "Output"] }, Open ]] }, 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[TextData[{ StyleBox["a)", FontSlant->"Italic"], " ", StyleBox["H[s]", "Input"], " is the product of ", StyleBox["F[s_] = 1/(s+1)^2", "Input"], " and ", StyleBox["G[s_] = s/(s^2+9)", "Input"], ". The inverse Laplace transforms of these two functions are (see p.300):" }], "Text"], Cell[TextData[{ StyleBox["f[t_] = E^(-t)t;", "Input"], " ", StyleBox["g[t_] = Cos[3t];", "Input"] }], "Input"], Cell[TextData[{ "By the Convolution Theorem, the inverse Laplace transform of ", StyleBox["H[s]", "Input"], " is:" }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox[ "h[t_] = Integrate[f[t-u]g[u],{u,0,t}]", "Input"]], "Input"], Cell[BoxData[ \(\(-\(1\/50\)\)\ E\^\(-t\)\ \((\(-4\) + 5\ t)\) + 1\/50\ \((\(-4\)\ Cos[3\ t] + 3\ Sin[3\ t])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "InverseLaplaceTransform[", StyleBox["s/((s+1)^2(s^2+9))", "Input"], ",s,t]" }], "Input"], Cell[BoxData[ \(\(2\ E\^\(-t\)\)\/25 - \(E\^\(-t\)\ t\)\/10 + 1\/50\ \((\(-4\)\ Cos[3\ t] + 3\ Sin[3\ t])\)\)], "Output"] }, Open ]], Cell[TextData[StyleBox["b)", FontSlant->"Italic"]], "Text"], Cell["Clear[t,y,g]", "Input", AspectRatioFixed->True], Cell["\<\ Y[s_] := LaplaceTransform[y[t],t,s]; G[s_] := LaplaceTransform[g[t],t,s];\ \>", "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ "eqn = LaplaceTransform[\n ", StyleBox["y''''[t] + 10y''[t] + 9y[t] == g[t]", "Input"], ",t,s]" }], "Input", AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{ \(9\ LaplaceTransform[y[t], t, s]\), "+", \(s\^4\ LaplaceTransform[y[t], t, s]\), "-", \(s\^3\ y[0]\), "+", RowBox[{"10", " ", RowBox[{"(", RowBox[{ \(s\^2\ LaplaceTransform[y[t], t, s]\), "-", \(s\ y[0]\), "-", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], ")"}]}], "-", RowBox[{\(s\^2\), " ", RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "[", "0", "]"}]}], "-", RowBox[{"s", " ", RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "[", "0", "]"}]}], "-", RowBox[{ SuperscriptBox["y", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "0", "]"}]}], "==", \(LaplaceTransform[g[t], t, s]\)}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ tran = Solve[eqn/.{y[0]->2, y'[0]->1, y''[0]->0, \ y'''[0]->0},Y[s]]\ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{LaplaceTransform[y[t], t, s] \[Rule] \(-\(\(\(-10\) - 20\ s - s\^2 - 2\ s\^3 - LaplaceTransform[g[t], t, s]\)\/\(9 + 10\ s\^2 + s\^4\)\)\)}}\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["soln = InverseLaplaceTransform[tran,s,t]", "Input"], Cell[BoxData[ \({{y[t] \[Rule] 20\ \((Cos[t]\/8 - 1\/8\ Cos[3\ t])\) + 2\ \((\(-\(Cos[t]\/8\)\) + 9\/8\ Cos[3\ t])\) + \[Integral]\_0\%t \( g[t1]\ \((1\/8\ Sin[t - t1] - 1\/24\ Sin[3\ \((t - t1)\)])\)\) \[DifferentialD]t1 - Sin[t]\/8 + 10\ \((Sin[t]\/8 - 1\/24\ Sin[3\ t])\) + 3\/8\ Sin[3\ t]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y[t] /. First[soln/.g[t1]->Cos[t1]] //Simplify", "Input"], Cell[BoxData[ \(1\/192\ \((429\ Cos[t] - 45\ Cos[3\ t] + 4\ \((52 + 3\ t - 4\ Cos[2\ t])\)\ Sin[t])\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y[t] /. First[soln/.g[t1]->Exp[t1]] //Simplify", "Input"], Cell[BoxData[ \(1\/80\ \((4\ E\^t + 175\ Cos[t] - 19\ Cos[3\ t] + 85\ Sin[t] - 3\ Sin[3\ t]) \)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["y[t] /. First[soln/.g[t1]->t1^2]", "Input"], Cell[BoxData[ \(\(-\(20\/81\)\) + t\^2\/9 + Cos[t]\/4 + 20\ \((Cos[t]\/8 - 1\/8\ Cos[3\ t])\) - 1\/324\ Cos[3\ t] + 2\ \((\(-\(Cos[t]\/8\)\) + 9\/8\ Cos[3\ t])\) - Sin[t]\/8 + 10\ \((Sin[t]\/8 - 1\/24\ Sin[3\ t])\) + 3\/8\ Sin[3\ t]\)], "Output"] }, Open ]] }, 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[TextData[StyleBox["a)", FontSlant->"Italic"]], "Text"], Cell[TextData[StyleBox[ "A = {{ 7,-3, 3},\n\t {-1, 5, 1},\n\t { 2,-2, 8}};", "Input"]], "Input"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Det[A - r*IdentityMatrix[3]] == 0", FontFamily->"Courier", FontWeight->"Bold"]], "Input"], Cell[BoxData[ \(240 - 124\ r + 20\ r\^2 - r\^3 == 0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[%]", "Input"], Cell[BoxData[ \({{r \[Rule] 4}, {r \[Rule] 6}, {r \[Rule] 10}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ eqns = (A-r*IdentityMatrix[3]).{x1,x2,x3} == {0,0,0} \ //Simplify\ \>", "Input"], Cell[BoxData[ \({7\ x1 - r\ x1 - 3\ x2 + 3\ x3, \(-x1\) + 5\ x2 - r\ x2 + x3, 2\ x1 - 2\ x2 + 8\ x3 - r\ x3} == {0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[eqns/.r->4]", "Input"], Cell[BoxData[ \({{x1 \[Rule] x2, x3 \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "An eigenvector for ", StyleBox["r == 4", "Input"], " is ", StyleBox["v1 = {1,1,0}", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["Solve[eqns/.r->6]", "Input"], Cell[BoxData[ \({{x1 \[Rule] 0, x2 \[Rule] x3}}\)], "Output"] }, Open ]], Cell[TextData[{ "An eigenvector for ", StyleBox["r == 6", "Input"], " is ", StyleBox["v1 = {0,1,1}", "Input"], "." }], "Text"], Cell[CellGroupData[{ Cell["Solve[eqns/.r->10]", "Input"], Cell[BoxData[ \({{x1 \[Rule] x3, x2 \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "An eigenvector for ", StyleBox["r == 10", "Input"], " is ", StyleBox["v1 = {1,0,1}", "Input"], "." }], "Text"], Cell[TextData[{ "Now we try ", StyleBox["Eigensystem[]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["?Eigensystem", "Input", AspectRatioFixed->True], Cell[BoxData[ \("Eigensystem[m] gives a list {values, vectors} of the eigenvalues and \ eigenvectors of the square matrix m."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell["Eigensystem[A]", "Input", AspectRatioFixed->True], Cell[BoxData[ \({{4, 6, 10}, {{1, 1, 0}, {0, 1, 1}, {1, 0, 1}}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["b)", FontSlant->"Italic"], " The columns of ", StyleBox["T", "Input"], " are the eigenvectors of ", StyleBox["A", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell["\<\ T = Transpose[{{1,1,0},{0,1,1},{1,0,1}}]; 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We can see directly that they are dependent at the two points ", StyleBox["t==", FontFamily->"Courier", FontWeight->"Bold"], " ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["1", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["{X1[0],X2[0],X3[0]}//MatrixForm", "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"0", "0", "0"}, {"0", "0", "1"}, {"0", "1", "0"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["{X1[1],X2[1],X3[1]}//MatrixForm", "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {"1", "1", "1"}, {"1", "1", "1"}, {"1", "1", "1"} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "So, for example, ", StyleBox["1*X1[0] + 0*X2[0] + 0*X3[0] == {0,0,0}", "Input"], ", and\n", StyleBox["1*X1[1] - 1*X2[1] + 0*X3[1] == {0,0,0}", "Input"], "." }], "Text"], Cell[TextData[{ "As ", StyleBox["vector functions", FontSlant->"Italic"], ", ", StyleBox["X1[t]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2[t]", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["X3[t]", FontFamily->"Courier", FontWeight->"Bold"], " are still linearly independent on any interval. The only linear \ combination of them that equals ", StyleBox["{0,0,0}", "Input"], " for ", StyleBox["all", FontSlant->"Italic"], " values of ", StyleBox["t", "Input"], " is when all coefficients are zero. To see this, we'll try to solve just \ three of the infinite number (", StyleBox["all", FontSlant->"Italic"], " values of ", StyleBox["t", "Input"], ") of equations the coefficients must satisfy:" }], "Text"], Cell[CellGroupData[{ Cell["\<\ eqns = {c1 X1[0] + c2 X2[0] + c3 X3[0] == {0,0,0}, c1 X1[1] + c2 X2[1] + c3 X3[1] == {0,0,0}, c1 X1[2] + c2 X2[2] + c3 X3[2] == {0,0,0}}\ \>", "Input"], Cell[BoxData[ \({{0, c3, c2} == {0, 0, 0}, {c1 + c2 + c3, c1 + c2 + c3, c1 + c2 + c3} == {0, 0, 0}, {2\ c1 + 2\ c2 + 4\ c3, 4\ c1 + 4\ c2 + c3, 8\ c1 + c2 + 2\ c3} == {0, 0, 0}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Solve[eqns]", "Input"], Cell[BoxData[ \({{c2 \[Rule] 0, c3 \[Rule] 0, c1 \[Rule] 0}}\)], "Output"] }, Open ]], Cell[TextData[{ "We could also argue that a polynomial in ", StyleBox["t", "Input"], " is identically zero if and only if all of its coefficients are ", StyleBox["0", "Input"], ":" }], "Text"], Cell[CellGroupData[{ Cell["c1 X1[t] + c2 X2[t] + c3 X3[t] == {0,0,0}", "Input"], Cell[BoxData[ \({c1\ t + c2\ t + c3\ t\^2, c3 + c1\ t\^2 + c2\ t\^2, c2 + c3\ t + c1\ t\^3} == {0, 0, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "For example, the third coordinate ", StyleBox["c2 + c3 t + c1 t^3 == 0", "Input"], " for all ", StyleBox["t", "Input"], " only if ", StyleBox["c1 == c2 == c3 == 0", "Input"], "." }], "Text"], Cell[TextData[{ StyleBox["c)", FontSlant->"Italic"], " The coefficients in the system of homogeneous differential equations \ satisfied by ", StyleBox["X1[t]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X2[t]", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["X3[t]", FontFamily->"Courier", FontWeight->"Bold"], " must be discontinuous at the points ", StyleBox["t ==", FontFamily->"Courier", FontWeight->"Bold"], " ", StyleBox["0", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["1", FontFamily->"Courier", FontWeight->"Bold"], ". 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