(*********************************************************************** 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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There we find, for example, that the option Contours->{a,b,c,...} \ will graph the contours with specific values of the constant. Suppose the solution is given in the form g(x,y) = c. We can then use the \ following model, with suitable editing for each example.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["g[x_,y_]=?"], "Input", AspectRatioFixed->True], Cell[TextData[ "curves=ContourPlot[g[x,y],{x,-3,3},{y,-3,3},\n\t\t\t\t\t\ Axes->True,Contours->15,\n\t\t\t\t\tContourShading->False,\n\t\t\t\t\t\ ContourSmoothing->True,\n\t\t\t\t\tPlotPoints->25]"], "Input", AspectRatioFixed->True], Cell[TextData[{ "Example 1. ", StyleBox["The equation ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(y' = x\^2\), "TraditionalForm"], "/", \((1 - y\^2\)}], TraditionalForm]]], StyleBox[") on page 34 of the text has solution ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`x\^3\)]], StyleBox["+ 3y -", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(\ y\^3\ \)\)]], StyleBox["= c.", FontWeight->"Plain"] }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(g[x_, y_] = x\^3 - 3 y + y\^3\)], "Input"], Cell["\<\ curves=ContourPlot[g[x,y],{x,-3,3},{y,-3,3}, \t\t\t\t\tAxes->True,Contours->25, \t\t\t\t\tContourShading->False, \t\t\t\t\tContourSmoothing->True, \t\t\t\t\tPlotPoints->25]\ \>", "Input", AspectRatioFixed->True], Cell["\<\ Note that the denominator of the right hand side of the equation \ f(t, y) is zero when y is 1 or -1, and that all the solution curves have a \ vertical when y is 1 or -1. \ \>", "Text"], Cell["\<\ If we wanted the solution curve for which y = 2 when x = 1 , we set \ c = g(1,2) \ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(g[x_, y_] = x\^3 - 3 y + y\^3\)], "Input"], Cell["\<\ curves=ContourPlot[g[x,y],{x,-3,3},{y,-3,3}, \t\t\t\t\tAxes->True,Contours->{g[1,2]}, \t\t\t\t\tContourShading->False, \t\t\t\t\tContourSmoothing->True, \t\t\t\t\tPlotPoints->25]\ \>", "Input", AspectRatioFixed->True], Cell["\<\ The domain of the solution with y = 2 when x = 1 can be roughly \ found by going to the curve at the point (1,2) and seeing how far you can go \ to the left and right and still have a well-defined function of x, in this \ case from about -2.5 to about 1.35.\ \>", "Text"], Cell[TextData[{ "Example 2. ", StyleBox[" The equation y' = (y cos x)/(1+2", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`y\^2\)]], StyleBox[") on page 36 has solution\n ln |y| ", FontWeight->"Plain"], Cell[BoxData[ \(TraditionalForm\`\(+y\^2\)\)]], StyleBox[" = sin x + C.", FontWeight->"Plain"], " " }], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(g[x_, y_] = Log[Abs[y]] + y\^2 - Sin[x]\)], "Input"], Cell[CellGroupData[{ Cell["\<\ curves=ContourPlot[g[x,y],{x,-3,3},{y,.1,3}, \t\t\t\t\tAxes->True,Contours->15, \t\t\t\t\tContourShading->False, \t\t\t\t\tContourSmoothing->True, \t\t\t\t\tPlotPoints->25]\ \>", "Input", AspectRatioFixed->True], Cell[TextData["The solution which has y=1 when x=0 is given by"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(g[x_, y_] = Log[Abs[y]] + y\^2 - Sin[x] - 1\)], "Input"], Cell["\<\ curves=ContourPlot[g[x,y],{x,-3,3},{y,.1,3}, \t\t\t\t\tAxes->True,Contours->{g[0,1]}, \t\t\t\t\tContourShading->False, \t\t\t\t\tContourSmoothing->True, \t\t\t\t\tPlotPoints->25]\ \>", "Input", AspectRatioFixed->True] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, ScreenStyleEnvironment->"Presentation", WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{622, 570}, WindowMargins->{{199, Automatic}, {Automatic, 152}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding" ] (*********************************************************************** 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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