Series Methods Reference

Differential Equations · Axiom Academy

FORMULA SHEET Series Methods Reference Reference for power series and Frobenius methods, plus the special functions — Bessel, Legendre, and Airy — that solve them Valid at an ordinary point x_0 — assume a series solution, differentiate term-by-term, substitute, and match coefficients. For the standard-form equation below, classify x_0 by the analyticity of P(x) and Q(x) . For a regular singular point at x = 0 , extend the power series with a leading power x^r . Write the DE in the form below, so p(x) = (x-x_0)P(x) and q(x) = (x-x_0)^2Q(x) are analytic at x_0=0 . The relationship between the two indicial roots r_1, r_2 ( ) determines the form of the second solution. Legendre Polynomial Properties The shape a recurrence takes after matching coefficients, from simplest to most general.

This is the written version of the interactive lesson above. See the full Differential Equations course.