power-series method
Many differential equations have no solution writable with elementary functions, yet physics still needs the answer. The power-series method says: do not look for a formula, build the answer one coefficient at a time. You assume the solution is an infinite polynomial — a power series — substitute it into the equation, and let the equation itself dictate every coefficient. It is the workhorse for solving a linear ODE near an ordinary point.
The recipe is mechanical. Write y = sum over k>=0 of a_k (x - x0)^k about an ordinary point x0. Differentiate term by term to get y' and y'', substitute all three into the equation, collect terms by power of (x - x0), and demand that the coefficient of each power vanish (since the whole sum must be identically zero). This yields a recurrence relation that expresses each a_k in terms of earlier coefficients. You are free to choose a_0 and a_1 (the two arbitrary constants of a second-order equation, fixed by initial conditions); every later coefficient then follows by cranking the recurrence. The result is two independent solutions as explicit, convergent series.
This is how Airy's equation y'' = x y, the simple harmonic equation, and the analytic parts of Legendre, Hermite and Chebyshev are actually solved before anyone names a polynomial. Its strength is generality and honesty: it always works at an ordinary point, the series converges, and truncating it gives a usable numerical approximation with controllable error. Its limitation is that the series is only guaranteed near x0 (out to the nearest singularity), and it does not by itself handle singular points — for those you upgrade to the method of Frobenius.
Solve y'' + y = 0 by series. Put y = sum a_k x^k; substituting gives the recurrence a_{k+2} = -a_k/((k+2)(k+1)). Choosing a_0 = 1, a_1 = 0 builds 1 - x^2/2! + x^4/4! - ... = cos x; choosing a_0 = 0, a_1 = 1 builds x - x^3/3! + ... = sin x. The method reconstructs the familiar trig functions from nothing but the equation.
The two free constants a_0, a_1 generate the two independent solutions.
Index-shifting is where most errors hide: when you write y'' as a series, re-index so every term carries the same power (x - x0)^k before you equate coefficients, or you will collect the wrong terms.