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Notebook[{

Cell[CellGroupData[{
Cell["Homogeneous Systems of Ordinary Differential Equations", "Subtitle"],

Cell[CellGroupData[{

Cell["Linear Systems", "Section"],

Cell[CellGroupData[{

Cell["Example 1 - Solving from the Eigensystem", "Subsection"],

Cell["\<\
Suppose that you have the system of equations x'(t) = -y(t) and y'(t) = \
-x(t). If you first find the eigensystem of the coefficient matrix, you can \
write the general solution from it.\
\>", "Text"],

Cell[BoxData[
    RowBox[{
      RowBox[{"Eigensystem", "[", 
        RowBox[{"(", "\[NegativeThinSpace]", GridBox[{
              {"0", \(-1\)},
              {\(-1\), "0"}
              }], "\[NegativeThinSpace]", ")"}], "]"}], "//", 
      "MatrixForm"}]], "Input"],

Cell["\<\
The following set of commands will detemine the solution for various values \
of the arbitrary constants and then plot those solutions as in a phase \
diagram (parametric plot).\
\>", "Text"],

Cell[BoxData[{
    \(Off[General::spell]\ \), "\[IndentingNewLine]", 
    \(Off[General::spell1]\), "\[IndentingNewLine]", 
    \(\(x[t_] = c1\ Exp[\(-t\)] + c2\ Exp[t];\)\), "\[IndentingNewLine]", 
    \(y[t_] = c1\ Exp[\(-t\)] - c2\ Exp[t]; 
    fncarray = 
      Table[{x[t], y[t]} /. {c1 \[Rule] i, c2 \[Rule] j}, {i, \(-6\), 6, 
          2}, {j, \(-6\), 6, 2}]; 
    tograph = Flatten[fncarray, 1];\), "\[IndentingNewLine]", 
    \(Short[tograph, 2]\), "\[IndentingNewLine]", 
    \(\(graphone = 
        ParametricPlot[Evaluate[tograph], {t, \(-2\), 2}, 
          PlotRange \[Rule] {{\(-15\), 15}, {\(-15\), 15}}];\)\)}], "Input"],

Cell["\<\
We can also look at a vector field plot of the functions, by plotting arrows \
at points where the arrows are specified by the x' (horizontal component) and \
y' (vertical component)values. These will therefore be in the dy/dx \
directions. You must begin by reading in the following graphics package.\
\>", "Text"],

Cell[BoxData[
    \(<< Graphics`PlotField`\)], "Input"],

Cell[BoxData[
    \(graphtwo = 
      PlotVectorField[{\(-y\), \(-x\)}, {x, \(-15\), 15}, {y, \(-15\), 15}, 
        ScaleFunction \[Rule] \((1 &)\), DisplayFunction \[Rule] Identity]; 
    Show[graphone, graphtwo, 
      PlotRange \[Rule] {{\(-15\), 15}, {\(-15\), 15}}, 
      AspectRatio \[Rule] 1];\)], "Input"],

Cell["\<\
Note how the arrows line up with the solution curves.
What is happening to the solutions as t -> \[Infinity] ? How does it depend \
on your starting point?\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 2 - Solving Using DSolve", "Subsection"],

Cell["\<\
Here is one where the eigenvalues are complex. This solution would take a \
little work on your part to write, based on the eigensystem. We will do this \
in class to verify the results. For this one, we will plot the solutions and \
vector fields together. The system of equations looks very much like the one \
above, but the solutions are quite different (as can be seen from the \
eigensystem): x'(t) = y(t) and y'(t) = -x(t).\
\>", "Text"],

Cell[BoxData[{
    \(MatrixForm[mat = {{0, 1}, {\(-1\), 0}}]\), "\n", 
    \(Eigensystem[mat] // MatrixForm\)}], "Input"],

Cell["\<\
We will let Mathematica solve this system of equations for now.\
\>", "Text"],

Cell[BoxData[{
    \(Clear[x, y, t]\), "\n", 
    \(solution = 
      DSolve[Thread[{\(x'\)[t], \(y'\)[t]} \[Equal] mat . {x[t], y[t]}], {x[
            t], y[t]}, t]\), "\[IndentingNewLine]", 
    \(solx := solution[\([1, 1, 2]\)]\), "\n", 
    \(soly := solution[\([1, 2, 2]\)]\)}], "Input"],

Cell["\<\
Now we will assign different values to the arbitrary constants.\
\>", "Text"],

Cell[BoxData[{
    \(fncarray = 
      Table[{solx, soly} /. {C[1] \[Rule] i, C[2] \[Rule] j}, {i, \(-6\), 6, 
          2}, {j, \(-6\), 6, 2}]; tograph = Flatten[fncarray, 1]; 
    Short[tograph, 2];\), "\[IndentingNewLine]", 
    \(\(graphthree = 
        ParametricPlot[Evaluate[tograph], {t, \(-1\), 1}, 
          PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
          AspectRatio \[Rule] 1, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(graphdf = 
        PlotVectorField[{y, \(-x\)}, {x, \(-10\), 10}, {y, \(-10\), 10}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(Show[graphthree, graphdf, 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"],

Cell["\<\
What is happening to these solutions as t\[Rule]\[Infinity]?\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 3 - Another Way to Call on Your Solutions", "Subsection"],

Cell[BoxData[{\(Clear[x, y, sol, mat, t]\), "\[IndentingNewLine]", 
    RowBox[{"MatrixForm", "[", 
      RowBox[{"mat", "=", 
        RowBox[{"(", GridBox[{
              {\(-5\), "3"},
              {\(-2\), \(-10\)}
              }], ")"}]}], "]"}], "\[IndentingNewLine]", \(Eigensystem[mat] // 
      MatrixForm\), "\n", \(sol = 
      DSolve[Thread[{\(x'\)[t], \(y'\)[t]} \[Equal] mat . {x[t], y[t]}], {x[
            t], y[t]}, t]\)}], "Input"],

Cell[BoxData[{
    \(funs = 
      Table[\({x[t], y[t]} /. sol\) /. {C[1] \[Rule] i, 
            C[2] \[Rule] j}, {i, \(-6\), 6, 3}, {j, \(-6\), 6, 3}]; 
    toplot = Flatten[funs, 1]; Length[toplot];\), "\[IndentingNewLine]", 
    \(\(pptwo = 
        ParametricPlot[Evaluate[toplot], {t, \(-2\), 1}, 
          Compiled \[Rule] False, 
          PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(graph4 = 
        PlotVectorField[mat . {x, y}, {x, \(-10\), 10}, {y, \(-10\), 10}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(Show[pptwo, graph4, PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
        AspectRatio \[Rule] 1, \ 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"],

Cell["\<\
What is happening to these solutions as t\[Rule]\[Infinity]?\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 4 - Zero Eigenvalue", "Subsection"],

Cell[BoxData[{
    \(Clear[x, y, sol, mat, t]\), "\[IndentingNewLine]", 
    \(MatrixForm[mat = {{\(-4\), 6}, {\(-2\), 3}}]\), "\[IndentingNewLine]", 
    \(Eigensystem[mat] // MatrixForm\), "\[IndentingNewLine]", 
    \(sol = 
      DSolve[Thread[{\(x'\)[t], \(y'\)[t]} \[Equal] mat . {x[t], y[t]}], {x[
            t], y[t]}, t]\)}], "Input"],

Cell[BoxData[{
    \(funs = 
      Table[\({x[t], y[t]} /. sol\) /. {C[1] \[Rule] i, 
            C[2] \[Rule] j}, {i, \(-6\), 6, 3}, {j, \(-6\), 6, 3}]; 
    toplot = Flatten[funs, 1]; Length[toplot];\), "\[IndentingNewLine]", 
    \(\(pptwo = 
        ParametricPlot[Evaluate[toplot], {t, \(-2\), 1}, 
          Compiled \[Rule] False, 
          PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(graph4 = 
        PlotVectorField[mat . {x, y}, {x, \(-10\), 10}, {y, \(-10\), 10}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(Show[pptwo, graph4, PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
        AspectRatio \[Rule] 1, \ 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"],

Cell["\<\
What is different about this compared to the problems above? What impact did \
the zero eigenvalue have on the solutions? Why?\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Example 5 - Repeated Root", "Subsection"],

Cell[BoxData[{
    \(Clear[x, y, sol, mat, t]\), "\[IndentingNewLine]", 
    \(MatrixForm[mat = {{2, 1}, {\(-1\), 4}}]\), "\[IndentingNewLine]", 
    \(Eigensystem[mat] // MatrixForm\), "\n", 
    \(sol = 
      DSolve[Thread[{\(x'\)[t], \(y'\)[t]} \[Equal] mat . {x[t], y[t]}], {x[
            t], y[t]}, t]\)}], "Input"],

Cell[BoxData[{
    \(funs = 
      Table[\({x[t], y[t]} /. sol\) /. {C[1] \[Rule] i, 
            C[2] \[Rule] j}, {i, \(-6\), 6, 3}, {j, \(-6\), 6, 3}]; 
    toplot = Flatten[funs, 1]; Length[toplot];\), "\[IndentingNewLine]", 
    \(\(pptwo = 
        ParametricPlot[Evaluate[toplot], {t, \(-2\), 1}, 
          Compiled \[Rule] False, 
          PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(graph4 = 
        PlotVectorField[mat . {x, y}, {x, \(-10\), 10}, {y, \(-10\), 10}, 
          DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", 
    \(\(Show[pptwo, graph4, PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}, 
        AspectRatio \[Rule] 1, \ 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"]
}, Closed]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Assignment", "Section"],

Cell[TextData[{
  "Try out two other systems of linear homogeneous differential equations - \
from your text or from your imagination - and see what happens. The code used \
in examples 3-4-5 will probably be the easiest to use, since all you will \
need to change is your coefficient matrix. Briefly describe what you tried \
and the results (no ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " handouts for this part - just YOUR words). Tell me what the visualization \
of the vector field and the set of solutions does to enhance your \
understanding of the nature of the solutions. \nWrite two paragraphs telling \
me what the eigenvalues of the coefficient matrix tell you about the \
stability of the equilibrium point (0,0)? \n",
  StyleBox["As a final page, printout one of your plots (a different one for \
each student - so don't use my examples) showing a set of solutions where \
your eigenvectors were real (make sure your arrows go in the same direction \
as your solutions) and draw your eigenvectors on that graph in some different \
color.",
    FontWeight->"Bold"]
}], "Text"]
}, Closed]]
}, Open  ]]
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