Power Series Solutions

A rigorous coefficient-matching method for local power-series solutions near ordinary points.

A sequence tracks terms; a series tracks their accumulating partial sums. Indexed terms appear along a horizontal progression above partial-sum markers. Shrinking term arrows and a target band distinguish terms approaching zero from partial sums actually converging.

Precise definition

A power-series solution assumes $y(x)=\sum_{n=0}^\infty a_n(x-x_0)^n$ and differentiates term-by-term within its radius of convergence. Substitution into a linear ODE and equality of coefficients produce a recurrence for $a_n$. An ordinary point is one where the normalized coefficient functions are analytic.

Notation and mathematical language

$y'=\sum_{n=0}^\infty(n+1)a_{n+1}(x-x_0)^n$ and $y''=\sum_{n=0}^\infty(n+2)(n+1)a_{n+2}(x-x_0)^n$. Reindex all sums to the same power before equating coefficients. Initial values determine $a_0=y(x_0)$ and $a_1=y'(x_0)$.

Conceptual picture

The method replaces differentiation by algebra on an infinite coefficient sequence. Even and odd coefficients often decouple, revealing two independent solutions for a second-order equation. The nearest singularity of the coefficient functions often limits the convergence disk.

Fully worked example

Interpretation and application

Series methods solve variable-coefficient equations without elementary closed forms and provide local approximations. Numerical use requires choosing a truncation and bounding or estimating the tail; symbolic equality refers to the infinite convergent series.