Series solutions to differential equations can be grubby or elegant, depending on your perspective.
Power series solutions
At one level, there’s nothing profound going on. To find a series solution to a differential equation, assume the solution has a power series, stick the series into the equation, and solve for the coefficients. The mechanics are tedious, but you just follow your nose.
But why should this work? Why should the solution have a power series? And why should it be possible to find the coefficients? After all, there are infinitely many of them!
As for why the solution should have a power series, sometimes it doesn’t. But there is a complete theory to tell you when the solution should have a power series or not. And when it doesn’t have a power series, it may have more general series solution.
The tedious steps in the middle of the solution process may obscure what’s happening between the first and last step; the hard part in the middle grabs our attention. But the big picture is that series solution process transforms a differential equation for a function y(x) into a recurrence relation for the coefficients in the power series for y. We turn a continuous problem into a discrete problem, a hard problem into an easy problem.
To look at an example, consider Airy’s equation:
y” – xy = 0.
I won’t go into all the details of the solution process—my claim above is that these details obscure the transformation I want to emphasize—but skip from the first to the last step.
If you assume y has a power series at 0 with coefficients an, then we find that the coefficients satisfy the recurrence relation
(n + 2)(n + 1)an+2 = an-1
for n ≥ 1. Initial conditions determine a0 and a1, and it turns out a2 must be 0. The recurrence relation shows how these three coefficients determine all the other coefficients.
Holonomic functions and sequences
There’s something interesting going on here, a sort of functor mapping differential equations to recurrence relations. If we restrict ourselves to differential equations and recurrence relations with polynomial coefficients, we get into the realm of holonomic functions.
A holonomic function is one that satisfies a linear differential equation with polynomial coefficients. A holonomic sequence is a sequence that satisfies a linear recurrence relation with polynomial coefficients. Holonomic functions are the generating functions for holonomic sequences.
Holonomic functions are interesting for a variety of reasons. They’re a large enough class of functions to include many functions of interest, like most special functions from analysis and a lot of generating functions from combinatorics. And yet they’re a small enough class to have nice properties, such as closure under addition and multiplication.
Holonomic functions are determined by a finite list of numbers, the coefficients of the polynomials in their defining differential equation. This means they’re really useful in computer algebra because common operations ultimately map one finite set of numbers to another.