Solving "Impossible" Differential Equations with Mathematica

(continued from last page...)

We started out by trying to solve:

This initial value problem is unsolvable by our standard techniques. So we asked Mathematica for a solution found numerically with the NDSolve command. After a little massaging we produced the answer:

This solution is only valid on the interval [0, 5] as specified both by us when we asked for the solution, and by Mathematica in the output shown above. We've never seen a solution to an initial value problem that looked quite like this before—we're used to seeing solutions that are just ordinary formulas, so we find ourselves wondering what good a solution that looks like the one above can possibly do for us?

To answer this question we need to ask ourselves another question: what do we expect to be able to do with an ordinary, analytic solution? If we can do the same things with this numerical solution that we usually can do with an analytic solution then the numerical solution might be quite satisfactory.

So what did you answer to the question? Some common answers might be among the following:

• To be able to substitute values into the solution function in order to find its value at particular points of interest. (e.g. in a population model the solution function giving the population size as a function of time might be used to predict the size of the population in 50 years time.)

• To be able to graph the solution to get a visual feel for any trends, etc., that might be present.

• To be able to stand back and admire the beauty of the formula we have found.

We will find that the solution Mathematica has found for us will be able to be used for both of the first two purposes. It fails, of course, on the last of our requirements, much to the chagrin of the pure mathematicians among us.

So let's see this approximate, numerical solution in action...