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Cell[TextData[{
  "Unfortunately, ",
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    FontSlant->"Italic"],
  " doesn't include much in the line of nice point-and-click interfaces for \
drawing solutions to the dynamical system defined by a vector field.  Here's \
how to do it through function calls."
}], "Text"],

Cell["Let's say your dynamical system is:", "Text"],

Cell[BoxData[{
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Cell["Let's define a rule table that encodes this system:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dynamicalSystem = {\(x'\)[t_] \[Rule] 
          x[t] \((3 - x[t] - 2  y[t])\), \[IndentingNewLine]\(y'\)[
            t_] \[Rule] y[t] \((2 - x[t] - y[t])\)}\)], "Input",
  CellLabel->"In[4]:="],

Cell[BoxData[
    RowBox[{"{", 
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          "\[Rule]", \(x[t]\ \((3 - x[t] - 2\ y[t])\)\)}], ",", 
        RowBox[{
          RowBox[{
            SuperscriptBox["y", "\[Prime]",
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Cell[TextData[{
  "Note that I've defined rules that replace ",
  Cell[BoxData[
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  " where the ",
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  Cell[BoxData[
      \(TraditionalForm\`\(x'\)[anything]\)]],
  ", not just ",
  Cell[BoxData[
      \(TraditionalForm\`\(x'\)[t]\)]],
  ".  By using a rule table of this form, we can do lots of interesting \
things.  For example, let's say we want to plot the vector field:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

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Cell[TextData[{
  "If having expressions like ",
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  " and such in your vector field bothers you, you can also get rid of them \
with this rule:"
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Yes, you can make the name of the function a pattern to.  Just so \
you can see what that is doing, consider:\
\>", "Text"],

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So I can now give this command, which is pretty much the same as my \
earlier command but it doesn't look quite so odd, and this time I'll save the \
result to be used in later pictures:\
\>", "Text"],

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