Mathematics & Science
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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:
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... |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker ©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |