Mathematics & Science
Solving "Impossible" Differential Equations
We have learned theorems that guarantee to us that under the right conditions an initial value problem has a solution that both exists, and is unique on some interval containing the x-value from the initial condition(s). However, we have also seen that despite this guarantee, it is frequently impossible to actually find this solution by standard techniques.
What this usually means is that the solution we seek cannot be written in a form that uses a combination of the standard "elementary" mathematical functions that we are used to. This may come as a shock at first, but absolutely the same problem exists for the majority of functions you might sketch by randomly drawing a functional graph on a piece of paper. In fact we might state this fact even more strongly: The vast majority of functions from the univeral set of all possible functions cannot be described with a formula. It is not surprising then that so many differential equations have solutions that fit into this category.
So what then? Should we just banish these "impossible" differential equations from our consideration and pretend that they don't exist? That would be fine, but all too many of them are the very equations which describe "real life" physical phenomena. These are the differential equations we have to solve to model the heat flowing through a steel rod, or put a space shuttle into orbit. We can't just not solve them.
The somewhat unsatisfying truth is that we must find approximate numerical solutions to most actual applied differential equations. Now remember, these solutions will not be given by the kind of formulas we are used to. Rather, numerical solutions will be lists of points which lie, hopefully, close to the actual solution's curve.
Some numerical methods play "join the dots" with these points to mimic, as best they can, the behavior of the actual solution. One note about this kind of solution: it can't contain any arbitrary constants. We get one solution, not a family of solutions. This means that the problems we solve must be initial value problems.
Later in the course we will learn how some of these numerical methods actually work, but for now we'll let Mathematica's pre-written numerical solver do the approximating for us. Let's go and see how it works.
ODE Laboratories: A Sabbatical Project by Christopher A. Barker
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