What good is a DE once you have the solution?

In an undergraduate course on differential equations, you see an equation, you find a solution. Wax on, wax off. It would be reasonable to assume that the differential equation is simply an obstacle to overcome and that once you have the solution, the differential equation has done its job and can be discarded.

It would seem backward to ask whether a function satisfies a differential equation. It is backward, but that doesn’t mean it’s a bad idea. It can be very useful to know that a function you’re interested in satisfies a nice differential equation. (It’s trivial to make up differential equations for any function; the key is to discover a nice differential equation, a simple relationship between a function and its derivatives.)

The previous post gave an example of this. There we said that although Halley’s root-finding method converges faster than Newton’s method, it also requires one more function evaluation, namely the second derivative of the function whose root you’re finding. But when that function satisfies a second order differential equation, you can get the second derivative for free once you’ve evaluated the function and its derivative. Well, at a deep discount at least if not free.

In that post we gave the example of a Bessel function. Once you have evaluated a Bessel function and its first derivative at a point. you can calculate the second derivative in much less time that it took to evaluate the Bessel function itself.

Bessel functions are not unique in this regard. Hypergeometric functions are a large and useful class of functions, and these functions all satisfy Riemann’s differential equation.

All the classic orthogonal polynomials—Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, Hermite polynomials, etc.—satisfy a linear second order differential equation. For example, the Legendre polynomials Pn satisfy

(1 - x^2) y''(x) - 2 x y'(x) + \nu (\nu + 1) y = 0

And its not just Bessel functions, hypergeometric functions, and orthogonal polynomials. Most special functions satisfy a useful differential equation. More specifically, a linear second order differential equation.

So far the only reason given for wanting to know what differential equation a function satisfies is to speed up calculations: evaluate two functions and get a third one for cheap. But there are many other reasons to care what differential equation a function satisfies. For example, a common way to show that two functions are equal is to show that they satisfy the same differential equation with the same initial conditions. This is applying a uniqueness theorem for differential equations, but many other equations about differential equations could be useful to apply.

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