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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:"
}], "Text"],

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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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Cell[TextData[{
  "Back to our dynamical system.  Now suppose you want to plot solutions on \
top of the vector field.  To do that, we should use ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s built-in numerical solving functions.  Here's how to turn our dynamical \
system rule table into a system of equations that ",
  ButtonBox["NDSolve",
    ButtonStyle->"MasterIndexLink"],
  " can handle:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ode = {\(x'\)[
            t] \[Equal] \((\(x'\)[t] /. 
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            t] \[Equal] \((\(y'\)[t] /. dynamicalSystem)\)}\)], "Input",
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Cell[BoxData[
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          "\[Equal]", \(x[t]\ \((3 - x[t] - 2\ y[t])\)\)}], ",", 
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}, Open  ]],

Cell[TextData[{
  "We also have to add in initial conditions.  Those must be added to the \
ODE, so we use the ",
  ButtonBox["Join",
    ButtonStyle->"MasterIndexLink"],
  " function:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Join[{a, b, c}, {x, y, z}]\)], "Input",
  CellLabel->"In[11]:="],

Cell[BoxData[
    \({a, b, c, x, y, z}\)], "Output",
  CellLabel->"Out[11]="]
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    RowBox[{"{", 
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        RowBox[{
          RowBox[{
            SuperscriptBox["x", "\[Prime]",
              MultilineFunction->None], "[", "t", "]"}], 
          "\[Equal]", \(x[t]\ \((3 - x[t] - 2\ y[t])\)\)}], ",", 
        RowBox[{
          RowBox[{
            SuperscriptBox["y", "\[Prime]",
              MultilineFunction->None], "[", "t", "]"}], 
          "\[Equal]", \(\((2 - x[t] - y[t])\)\ y[t]\)}], 
        ",", \(x[0] \[Equal] 2\), ",", \(y[0] \[Equal] 2\)}], "}"}]], "Output",\

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}, Open  ]],

Cell[TextData[{
  "Here's how to use ",
  ButtonBox["NDSolve",
    ButtonStyle->"MasterIndexLink"],
  ":"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(numSol1 = 
      NDSolve[Join[
          ode, {x[0] \[Equal] 2, y[0] \[Equal] 2}], \[IndentingNewLine]{x, 
          y}, {t, 0, 5}]\)], "Input",
  CellLabel->"In[13]:="],

Cell[BoxData[
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              Editable->False]}], ",", 
          RowBox[{"y", "\[Rule]", 
            TagBox[\(InterpolatingFunction[{{0.`, 5.`}}, "<>"]\),
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              Editable->False]}]}], "}"}], "}"}]], "Output",
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}, Open  ]],

Cell["\<\
The return value is a list of rule tables that encode numerical \
solutions.  You can use them pretty much like you'd use any regular function. \
 It returns a list of rule tables in case it happens to find multiple \
solutions, which is rare.  Since we usually just want the first solution, \
it's best to use the command\
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  ", as in something that might be true or false, or something to be solved.  \
That's different from the single equal ",
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  " that are used to make definitions.  For various reasons, you can use \
single equal inside an expression, because that makes a definition and \
returns the value of the definition.  That leads to bugs where you use ",
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  " instead of ",
  Cell[BoxData[
      \(TraditionalForm\` \[Equal] \)]],
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error message.  Here's what can go wrong if you use ",
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}], "Text"],

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Cell["Hmm, let's make it square:", "Text"],

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Cell["\<\
Now if you wanted to plot a lot of solution curves, the procedure I \
just described would get tedious.  So, we should automate it.  First, let's \
define a function that creates a numerical solution starting at a given \
initial condition:\
\>", "Text"],

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      First[NDSolve[
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Cell[TextData[{
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until I've acutally specified the initial conditions.  Here's what happens if \
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Cell["So this makes a bunch of solution curves:", "Text"],

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  "Again, I use ",
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Cell["\<\
So now I can make a list of rule tables, each representing a \
different numerical solution.  And I can plot them all at once like \
this:\
\>", "Text"],

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And you can show the solutions on top of the vector field as \
follows:\
\>", "Text"],

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*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)