Series Solutions Summary

Differential Equations · Axiom Academy

SUMMARY Series Solutions Summary Recapping Unit 7: solving differential equations with power series when elementary functions run out — and meeting Bessel, Legendre, and Airy along the way. At an ordinary point, assume , substitute, and match coefficients to get a recurrence — the series converges at least out to the nearest singular point of the equation. Points are ordinary, regular singular, or irregular singular — that classification alone decides which method (power series or Frobenius) can even be attempted. At a regular singular point, the Frobenius form works, with r fixed by the indicial equation r(r-1)+p_0r+q_0=0 — its two roots split into three distinct cases, one of which forces a logarithm term. Bessel's equation and Legendre's equation are the two special equations you'll meet again and again — their solutions ( and P_n ) are as fundamental to circular/spherical problems as sine and cosine are to constant-coefficient ones. Series methods aren't a fallback for equations sine and e^x can't handle — they're the general-purpose machinery that produces the special functions physics and engineering actually run on. Core Concept Series Solutions at Ordinary Points When every coefficient of y''+P(x)y'+Q(x)y=0 is analytic at x_0 (an ordinary point ), assume this series form, differentiate term-by-term, shift indices so every sum lines up on the same power of x , then match coefficients. The two free constants a_0,a_1 generate two linearly independent solutions.

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