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Notebook[{

Cell[CellGroupData[{
Cell["Autonomous ODE's and Equilibrium States", "Subtitle",
  TextAlignment->Center],

Cell[TextData[{
  StyleBox[
  "The differential equations  y' =  r y and  y' = r y (1 - y/K) are both \
examples of ",
    FontSize->16],
  StyleBox["autonomous ",
    FontSize->16,
    FontSlant->"Italic"],
  StyleBox[
  "ODE's.  That is, the independent variable t does not appear (except \
implicitly through y) in the equation.  Consider the direction fields \
associated with these two equations\n(taking, for instance r = 1, K = 2):",
    FontSize->16]
}], "Text",
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  FontWeight->"Bold"],

Cell[BoxData[
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Cell[BoxData[
    \(\(PlotVectorField[{1, \ y}, {t, \(-3\), 3}, {y, \(-3\), 3}, 
      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",
  FontSize->16,
  FontWeight->"Bold"],

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]; \)\)], 
  "Input"],

Cell["\<\
Here there are two equilibrium solutions:  y = 0 and y = 2.  The
first is unstable, whereas the second is stable.  The general 
solution to   y' =  y (1 - y/2) is   y = 2 / (1 - C exp(-y)).  We
now graph this solution for C ranging from -5 to 5:\
\>", "Text",
  FontSize->16,
  FontWeight->"Bold"],

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["\<\
Sometimes an equilibrium arises which is neither stable nor
unstable.  Consider, for instance, y = 0 for the ODE 
y' = y^2 sin y:\
\>", "Text",
  FontSize->16,
  FontWeight->"Bold"],

Cell[BoxData[
    \(\(PlotVectorField[{1\ , \ y\ Sin[y]}, {t, \(-4\), 4}, {y, \(-4\), 4}, 
      Axes \[Rule] True, ScaleFunction \[Rule] \((1&)\), Ticks \[Rule] None, 
      Frame \[Rule] True, AspectRatio \[Rule] 1, \ PlotPoints\  -> \ 25]; 
    \)\)], "Input"],

Cell[TextData[StyleBox[
"Solutions starting near y = 0 are tend to converge/diverge from \ny=0 only \
fairly slowly.",
  FontWeight->"Bold"]], "Text",
  FontSize->16]
}, Open  ]]
},
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WindowMargins->{{155, Automatic}, {Automatic, 77}}
]


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Cached data follows.  If you edit this Notebook file directly, not using
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***********************************************************************)

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Cell[1731, 51, 84, 1, 64, "Subtitle"],
Cell[1818, 54, 508, 15, 144, "Text"],
Cell[2329, 71, 65, 1, 27, "Input"],
Cell[2397, 74, 224, 3, 59, "Input"],
Cell[2624, 79, 179, 5, 66, "Text"],
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}
]
*)




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