Math 235, Spring 2015

• Mondays: 14:00-15:30
• Wednesdays: 14:00-15:30

• Required: Course packet
• Recommended: Differential Equations, 3rd edition by Blanchard, Devaney, Hall

Both are available at the bookstore.

The Festival of Scholars is a campus-wide celebration of student work. It is an opportunity to discuss research, to exhibit, perform, or appreciate art, and to cross disciplinary boundaries. The Festival will be held on Friday, April 17, 2015. Classes will be cancelled on that day, but you are still required to participate in the Festival, either by presenting your work or attending presentations by your fellow students.

• If you have a Laptop, please install LaTeX ahead of time. See these resources.
• Mathematica: Tutorial 1, Tutorial 2
• A LaTeX template for reports can be found here.

Be sure to look at Exercise 4.7

• For the basic growth model :

  VectorPlot[{1, .5 P}, {t, 0, 5}, {P, -1, 1}]
  StreamPlot[{1, .5 P}, {t, 0, 5}, {P, -1, 1}]
  

• Other examples:

  StreamPlot[{1, .5 P (1 - P/10)}, {t, 0, 5}, {P, -2, 12}]
  StreamPlot[{1, 1 - y^2}, {t, 0, 5}, {y, -2, 2}]
  StreamPlot[{1, y (1 - y) Sin[t]}, {t, 0, 5}, {y, -2, 5}]
  

• Exam 1 is 13 February, during class time.
• Report 1 is due 16 February

• Plot the path for

  ParametricPlot[{Exp[t], Exp[2 t]},
       {t, -1, 1},
       PlotRange -> {0, 10}]
  

• Plot the vector field

  VectorPlot[{x + y, -x}, {x, -3, 3}, {y, -3, 3}]
  StreamPlot[{x + y, -x}, {x, -3, 3}, {y, -3, 3}]
  

Remember, SQRC closes early on Fridays

• with and :

  deltat = .05;
  f[y_] := Exp[y];
  values = RecurrenceTable[{
      t[k + 1] == t[k] + deltat,
      y[k + 1] == y[k] + deltat*f[y[k]],
      t[0] == 0, y[0] == 1
      }, {t, y}, {k, 0, 10}];
  Grid[values]
  ListLinePlot[values, PlotMarkers -> Automatic]
  

• with $y(0)=1$. We know solution is

  deltat = .05;
  n = 15;
  f[y_] := y^2;
  soln[t_] := 1/(1 - t);
  values = RecurrenceTable[{
      t[k + 1] == t[k] + deltat,
      y[k + 1] == y[k] + deltat*f[y[k]],
      t[0] == 0, y[0] == 1
      }, {t, y}, {k, 0, n}];
  Grid[values];
  eulerplot = ListLinePlot[values, PlotMarkers -> Automatic];
  actualplot = ListLinePlot[Table[{
       values[[i, 1]],
       soln[values[[i, 1]]]
       }, {i, 1, n + 1}], PlotStyle -> Purple];
  errorplot = ListLinePlot[Table[{
       values[[i, 1]],
       Abs[values[[i, 2]] - soln[values[[i, 1]]]]
       }, {i, 1, n}], PlotStyle -> Red];
  Show[eulerplot, actualplot]
  Show[eulerplot, actualplot, errorplot]
  

• See also this page.

Announcement: Report 2 due 20 March

Some comments about the first exam problem.

Report: Gravitation, due 20 April

Manipulate[Show[
  Plot[.5 x^2, {x, -4, 4}, PlotRange -> {0, 4},
   AspectRatio -> Automatic],
  StreamPlot[{Sqrt[2 H - x^2], -b (2 H - x^2)}, {x, -4, 4}, {H, 0,
    4}]], {b, 0, 3}]

and

Manipulate[Show[
  Plot[-Cos[x], {x, -5, 5}, PlotRange -> {-2, 3},
   AspectRatio -> Automatic],
  StreamPlot[{Sqrt[2 H + 2 Cos[x]], -b (2 H + 2 Cos[x])}, {x, -5,
    5}, {H, -2, 3}]], {b, 0, 1}]