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Notebook[{

Cell[CellGroupData[{
Cell["Autonomous ODE's and Equilibrium States", "Section"],

Cell["\<\
The differential equations y'=ry and y' = ry(1-y/K) are both
examples of autonomous ODE's.  That is, the independent variable
t does not appear in the equation.  Consider the direction fields 
associated with these two equations (taking, for instance, r=1,
and K=2):\
\>", "Text"],

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      Axes \[Rule] True, ScaleFunction \[Rule] \((1&)\), Ticks \[Rule] None, 
      Frame \[Rule] True, AspectRatio \[Rule] 1]; \)\)], "Input"],

Cell["\<\
The constant solution y=0 is distinguished.  Solutions that start
close to y=0 diverge rapidly away as t increases.  \
\>", "Text"],

Cell[BoxData[
    \(\(PlotVectorField[{1\ , \ y \((1 - y/2)\)}, {t, \(-3\), 3}, {y, \(-3\), 
        5}, Axes \[Rule] True, ScaleFunction \[Rule] \((1&)\), 
      Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1]; \)\)], 
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Cell[BoxData[{
    \(TextForm
    \`Here\ there\ are\ two\ equilibrium\ \(solutions : \ y\)\  = 
      \(0\ and\ y = 2. \ \ The\ first\)\ \), 
    \(TextForm\`is\ unstable, \ 
    but\ the\ second\ is\ stable . \ \ The\ general\ solution\ to\ \), 
    \(TextForm
    \`y' = \(y \((1 - y/2)\)\ \ is\ \ \ y\  = \ 
        2/\((1 - \ Ce\^\(-y\))\) . \ \ We\ now\ graph\ this\ function\)\), 
    \(TextForm\`for\ C\ ranging\ from\  - 5\ to\ \(5 : \)\)}], "Text"],

Cell[BoxData[
    \(Plot[
      Evaluate[Table\ [\ 2/\((1 - C\ Exp[\(-t\)])\)\ , {C, \(-5\), 5, 1}]], \n
      \t\t{t, \(-3\), 5}, \ AspectRatio -> 1, \ PlotRange\  -> \ {\(-3\), 5}]
      \)], "Input"],

Cell[BoxData[
    \(TextForm
    \`Sometimes\ an\ equilibrium\ arises\ which\ is\ neither\ stable\ nor\ 
      unstable . \nConsider, \ for\ instance, \ 
    y = \(0\ for\ the\ ODE\ \ y' = \ \((e\^y - \ 1)\) \((y + 1)\)\^2\)\)], 
  "Text"],

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      ScaleFunction \[Rule] \((1&)\), Ticks \[Rule] None, Frame \[Rule] True, 
      AspectRatio \[Rule] 1, \ PlotPoints\  -> \ 25]; \)\)], "Input"],

Cell["\<\
Solutions starting near y=0 tend to converge to or diverge from
y=0 only fairly slowly.\
\>", "Text"]
}, Open  ]]
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Cell[1792, 53, 290, 6, 135, "Text"],
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Cell[2153, 64, 224, 3, 87, "Input"],
Cell[2380, 69, 141, 3, 66, "Text"],
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