This file gives a very basic introduction to Maple for analyzing ordinary differential equations.

The first thing that you need to do is to open the DE toolbox.

** > with(DEtools); **

It is easy to get graphs of solution curves. This first example is a simple (first-order, linear, nonhomogeneous) equation:

** > eq:= diff(y(t),t) = sin(t)- 1/2*y(t); **

**
> DEplot(eq,[t,y],t=0..12,{[0,0]},y=-3..3,title=`Figure0`);
**

The solution curve starts out at the point (0,0). Notice also that it is a little jagged. This can be smoothed out by forcing Maple to use more points in the solution (the default is 20). You can do this by specifying the stepsize:

** >
DEplot(eq,[t,y],t=0..12,{[0,0]},y=-3..3,stepsize=0.1,title=`Figure0:
Smoother`); **

You can put several solutions on the same graph by specifying several initial points.

** > init:= seq([0,i],i=-3..3); **

All of the solution curves, no matter what the initial position, seem to approach one single sine curve.

You can get a better feel for the geometry behind the ODE by looking at a direction field. DEplot will produce this if you tell it to add arrows:

**
>DEplot(eq,[t,y],t=0..12,y=-3..3,arrows=THIN,title=`Figure2: A
Direction Field`);
**

You can get solutions on top of the direction field if you want to.

**
>DEplot(eq,[t,y],t=0..12,{[0,0]},stepsize=0.2,y=-3..3,arrows=THIN,
title=`Figure 3`);
**

Finally, you can also get formulas for the solution with the dsolve command.

**
> dsolve(eq,y(t));
**

Warning: There are not many ODEs that have simple formulae for their solutions. This example was linear, so there is a nice solution formula. If you try to use dsolve on more complicated problems, especially noninear problems, Maple may give no answer, a really strange formula, or lock up your work station. (Remember the first rule of Maple: Save Early, Save Often!) Use dsolve with caution.

Here is one more example; just for fun. This one is a second-order equation.

**
> eq2:= diff(y(t),t$2) + w^2*y(t) = sin(w*t);
**

© 1996 by Will Brother. All rights Reserved. File last modified on October 1, 1996.