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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[ 53808, 1949]*) (*NotebookOutlinePosition[ 54963, 1985]*) (* CellTagsIndexPosition[ 54919, 1981]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 325: Differential\tEquations Quiz 4 11/14/97", "Subsection", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell["<{{18, 126}, {Inherited, Inherited}}, PageBreakWithin->Automatic, InitializationCell->True, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ "Consider the non-linear initial value problem:\n", StyleBox[" x'[t] ==", FontFamily->"Courier"], StyleBox[" ", "Input", FontFamily->"Courier"], StyleBox["4x[t] - x[t]^2 - x[t] y[t]", "Input"], StyleBox["\n y'[t] ==", FontFamily->"Courier"], StyleBox[" ", "Input", FontFamily->"Courier"], StyleBox["y - 2x y - y^2", "Input"], StyleBox["\n", FontFamily->"Courier"], StyleBox["a) ", FontSlant->"Italic"], "Find all the critical points of this system.\n", StyleBox["b)", FontSlant->"Italic"], " Find the corrseponding linear system near each critical point and \ determine its eigenvalues.\n", StyleBox["c) ", FontSlant->"Italic"], "Discuss the type and stability of each critical point of the original \ system." }], "Subsubsection", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Solution"], "Subsubsection", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, PageBreakWithin->Automatic, CellLabelMargins->{{18, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell["\<\ F[x_,y_] = 4x - x^2 - x y; G[x_,y_] = y - 2x y - y^2;\ \>", "Input", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[TextData[{ StyleBox["a) ", FontSlant->"Italic"], "Find the critical points:" }], "Text", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["cp = Solve[{F[x,y] == 0, G[x,y] == 0}]", "Input", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ \({{x \[Rule] \(-3\), y \[Rule] 7}, {x \[Rule] 0, y \[Rule] 0}, { x \[Rule] 0, y \[Rule] 1}, {y \[Rule] 0, x \[Rule] 4}}\)], "Output"] }, Open ]], Cell[TextData[{ "The critical points are thus ", StyleBox["{-3,7}", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["{0,0}", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["{0,1}", FontFamily->"Courier", FontWeight->"Bold"], ", and ", StyleBox["{4,0}", FontFamily->"Courier", FontWeight->"Bold"], ".\n\n", StyleBox["b) and c) ", FontSlant->"Italic"], "To find the correspoinding linear systems and eigenvalues we need to \ examine the matrix of partial derivatives evaluated at each critical point." }], "Text", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ A = {{D[F[x,y],x], D[F[x,y],y]}, {D[G[x,y],x], D[G[x,y],y]}}; A//MatrixForm\ \>", "Input", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, AspectRatioFixed->True], Cell[BoxData[ TagBox[ RowBox[{"(", GridBox[{ {\(4 - 2\ x - y\), \(-x\)}, {\(\(-2\)\ y\), \(1 - 2\ x - 2\ y\)} }], ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]], Cell[TextData[{ "At the critical point ", StyleBox["{-3,7}", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", CellMargins->{{Inherited, 38}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["\<\ A1 = A /. cp[[1]]; 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