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                    Mathematica-Compatible Notebook

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(*CacheID: 232*)


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Notebook[{
Cell[BoxData[
    \(Needs["\<Graphics`PlotField`\>"]\)], "Input"],

Cell[BoxData[
    \(\(PlotVectorField[{1, 3  t^2/\((3  y^2 - 4)\)}, {t, 0, 
          3}, {y, \(-1.5\), 1.5}, \ Axes -> \ True, \n\t
        ScaleFunction \[Rule] \((1 &)\), Ticks \[Rule] None, 
        Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
        PlotPoints\  -> \ 15];\)\)], "Input"],

Cell["\<\
This is the direction field for the differential equation
in problem 13 on page 422.  Note that the equation can be
solved by hand to give that\
\>", "Text"],

Cell[BoxData[
    \(expr = \ y^3\  - 4  y\  - \ \(\(t\)\(^\)\(3\)\(\ \)\)\)], "Input"],

Cell["\<\
is equal to a constant.  So contour plots of this equation are \
solutions of the original
differential equation: \
\>", "Text"],

Cell[BoxData[
    \(ContourPlot[expr, {t, 0, 3}, {y, \(-1.5\), 1.5}, 
      ContourShading \[Rule] False]\)], "Input"],

Cell["\<\
In particular, \"expr\" has value -1 at the give initial point
(1,0).  We can look at just this contour:\
\>", "Text"],

Cell[BoxData[
    \(ContourPlot[expr, {t, 0, 3}, {y, \(-1.5\), 1.5}, 
      Contours \[Rule] {\(-1\)}, ContourShading \[Rule] False]\)], "Input"],

Cell["\<\
This is why Euler's method freaks out somewhere around 
t=1.6.  The domain of the solution simply ends around that
point.\
\>", "Text"]
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(***********************************************************************
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.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

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Cell[2734, 83, 145, 2, 43, "Input"],
Cell[2882, 87, 145, 4, 68, "Text"]
}
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*)




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End of Mathematica Notebook file.
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