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Cell[CellGroupData[{
Cell["\<\
Homogeneous Linear Systems of ODE's with constant \
coefficients\
\>", "Section"],

Cell[BoxData[{
    \(TextForm\`We\ consider\ systems\ of\ the\ form\), "\n", 
    \(TextForm\`x\&\[RightVector]' \((t)\)\  = \ 
      A\ \(x\&\[RightVector]\) \((t)\)\), "\n", 
    \(TextForm\`where\ \(x\&\[RightVector]\) \((t)\) = \ \((\(x\_1\) \((t)\), \
\  ... \ , \ \(x\_t\) \((t)\))\)\ is\ a\ vector\ whose\ components\ are\), "\n\
", 
    \(TextForm\`functions\ of\ t, \ 
    and\ A\ is\ an\ n\[Cross]
        n\ constant\ matrix . \ \ For\ practical\), "\n", 
    \(TextForm\`purposes, \ 
    we\ limit\ ourselves\ to\ the\ case\ n = 2. \)}], "Text"],

Cell[BoxData[
    \(RHS[x1_, x2_]\  := \ A . {x1, x2}\)], "Input"],

Cell[BoxData[
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Cell["Example 1 Source (A has two positive eigenvalues:", "Subsubsection"],

Cell["\<\
We define A in terms of its diagonalization--i.e. in terms of its \
eigenvalues and eigenvectors.  What, by the way, are its eigenvalues
and eigenvectors here?\
\>", "Text"],

Cell[BoxData[
    \(A\  = {{1, 1}, {1, \(-1\)}}\  . {{2, 0}, {0, 4}} . 
        Inverse[{{1, 1}, {1, \(-1\)}}]\)], "Input"],

Cell["\<\
Now we plot the direction field for A.  Unlike direction fields for \
single ODE's, direction fields for systems suppress the time variable.  This \
gives us an extra dimension for plotting the dependent variables, but it \
limits us to autonomous ODE's\
\>", "Text"],

Cell[BoxData[
    \(\(\(directions\)\(\ \)\(=\)\(\ \)\(RHS[x1, x2]\)\(\ \)\)\)], "Input"],

Cell[BoxData[
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      Axes -> \ True, \n\tScaleFunction \[Rule] \((1 &)\), 
      Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
      PlotPoints\  -> \ 15]\)], "Input"],

Cell[BoxData[
    \(\(\(\n\)\(rhs\  = \ RHS[x1[t], x2[t]]\)\)\)], "Input"],

Cell["\<\
Now we set up and solve the differentialequation, letting \"a\" and \
\"b\" be the initial values for
x1 and x2.\
\>", "Text"],

Cell[BoxData[
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          Part[rhs, 2], \n\t\ \ \ \ \ \ \ \ \ \ x1[0]\  == \ 
          a, \ \ \ \ \ \ \ \ \ x2[0]\  == \ b}\)], "Input"],

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          Table[\n\t\t\tsoln[
              t,  .01*Cos[angle],  .01*Sin[angle]], \n\t\t\t{angle, 0, 
              2*Pi,  .05*Pi}]], \n\t\ \ \ \ \ {t, \(-1.5\), 1.5}, 
        AspectRatio -> Automatic];\)\)], "Input"],

Cell["\<\
Note that there are two \"straight line\" solutions corresponding \
to the two eigenvectors of A and that other solutions tend to diverge from \
one of the lines and become parallel to the other.  Given any particular \
solution curve, the previous plot does not provide any
information about which way the solution moves as t increases.  On
the other hand you can figure this out by looking at the direction \
field.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Example 2: Saddle Point (A has real eigenvalues of opposite sign)\
\
\>", "Subsubsection"],

Cell["Same eigenvectors, different eigenvalues:", "Text"],

Cell[BoxData[
    \(A\  = {{1, 1}, {2, \(-2\)}}\  . {{ .6, 0}, {0, \(- .7\)}} . 
        Inverse[{{1, 1}, {2, \(-2\)}}]\)], "Input"],

Cell["The direction field:", "Text"],

Cell[BoxData[
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        Axes -> \ True, \n\tScaleFunction \[Rule] \((1 &)\), 
        Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
        PlotPoints\  -> \ 15];\)\)], "Input"],

Cell["The solution", "Text"],

Cell[BoxData[
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Cell[BoxData[
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          a, \ \ \ \ \ \ \ \ \ x2[0]\  == \ b}\)], "Input"],

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Cell[BoxData[
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          Table[\n\t\t\tsoln[t, r, 1], \n\t\t\t{r, \(-2\), 
              2,  .2}]], \n\t\ \ \ \ \ {t, \(-2\), 2}, 
        AspectRatio -> Automatic];\)\)], "Input"],

Cell["\<\
Can you guess what will happen if we change the
eigenvalues of A so that both are negative?\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example 3:  Indifferent point (Imaginary eigenvalues)", "Subsubsection"],

Cell["\<\
It's not so easy to specify a real matrix A with imaginary \
eigenvalues in terms of its diagonalization.  We take a slightly different \
approach.  Note that the middle matrix has eigenvalues \[PlusMinus]i.  Why \
should A have the same eigenvalues?\
\>", "Text"],

Cell[BoxData[
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        Inverse[{{1, 1}, {2, \(-1\)}}]\)], "Input"],

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        Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
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Cell["The solution", "Text"],

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          a, \ \ \ \ \ \ \ \ \ x2[0]\  == \ b}\)], "Input"],

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Cell[BoxData[
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          Table[\n\t\t\tsoln[t, r, 0], \n\t\t\t{r, 0, 
              2,  .1}]], \n\t\ \ \ \ \ {t, \(-5\), 5}, 
        AspectRatio -> Automatic];\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
Example  4: Spiral Point (A has complex conjugate \
eigenvalues)\
\>", "Subsubsection"],

Cell[" Note that the middle matrix has eigenvalues 1\[PlusMinus]i.  ", "Text"],

Cell[BoxData[
    \(A\  = {{1, 1}, {2, \(-2\)}}\  . {{1, 1}, {\(-1\), 1}} . 
        Inverse[{{1, 1}, {2, \(-2\)}}]\)], "Input"],

Cell[BoxData[
    \(\(\(directions\)\(\ \)\(=\)\(\ \)\(RHS[x1, x2]\)\(\ \)\)\)], "Input"],

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        Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
        PlotPoints\  -> \ 15];\)\)], "Input"],

Cell["The solution", "Text"],

Cell[BoxData[
    \(\(\(\n\)\(rhs\  = \ RHS[x1[t], x2[t]]\)\)\)], "Input"],

Cell[BoxData[
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          a, \ \ \ \ \ \ \ \ \ x2[0]\  == \ b}\)], "Input"],

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Cell[BoxData[
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Cell[BoxData[
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              t,  .01*Cos[angle],  .01*Sin[angle]], \n\t\t\t{angle, 0, 
              2*Pi,  .1*Pi}]], \n\t\ \ \ \ \ {t, \(-1.5\), 1.5}, 
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Cell["\<\
How would this have been different had the eigenvalues been -1\
\[PlusMinus]i?\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Example  5:  Improper node (A has repeated eigenvalues)", \
"Subsubsection"],

Cell["We start with the simplest possible example.", "Text"],

Cell[BoxData[
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        Ticks \[Rule] None, Frame \[Rule] True, AspectRatio \[Rule] 1, \ 
        PlotPoints\  -> \ 15];\)\)], "Input"],

Cell["The solution", "Text"],

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Cell[BoxData[
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              2*Pi,  .025*Pi}]], \n\t\ \ \ \ \ {t, \(-1.5\), 1.5}, 
        AspectRatio -> Automatic];\)\)], "Input"],

Cell["\<\
Note that solutions tend toward the same line regardless of whether
t goes to positive or negative infinity.  \
\>", "Text"]
}, Open  ]]
}, Open  ]]
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