the power-series method

Sometimes the coefficients in an equation are not constants but functions of x — say y'' + x y = 0 — and none of the tidy recipes (characteristic equations, undetermined coefficients) apply, because those all assume constant coefficients. The power-series method is a way out: instead of hunting for a solution built from familiar functions like e^(rt) or sin and cos, you build the solution out of an infinite polynomial, one coefficient at a time, and let the equation itself tell you what those coefficients must be.

Here is the idea walked through. You write the unknown as a power series y = sum from n=0 to infinity of a_n x^n = a_0 + a_1 x + a_2 x^2 + ..., where the a_n are numbers you do not yet know. Differentiating term by term gives series for y' and y''. You substitute all three series into the equation and collect everything by powers of x. Because a power series is zero only if every coefficient is zero, each power of x gives one equation that ties the unknown a_n together — a recurrence relation. Solving the recurrence expresses all the a_n in terms of a few free starting values (here a_0 and a_1, which turn out to be y(0) and y'(0)), and those free constants are exactly the arbitrary constants of the general solution.

This method is the workhorse for variable-coefficient equations, and it is how the great special functions of physics — Legendre, Bessel, Hermite — were first found. Two honest caveats: the series is only guaranteed to work cleanly when you expand about an ordinary point (where the coefficients are well-behaved), and the answer comes as an infinite series, not a formula in elementary functions. Often that series has no name and you simply keep as many terms as you need; sometimes, beautifully, it collapses into a polynomial or a known function.