Power Series Representations of DEQs

Uses Power Series

Method

To solve $a_2(x)y^{\prime \prime }+a_1(x)y^{\prime }+a_0(x)y=0$ where $y$ is analytic at $x_0$

1. Let $y=\sum _{n=0}^{\infty }{a}_n(x-x_0{)}^n$
2. Find $y^{\prime }$ and $y^{\prime \prime }$ and substitute them into the DEQ
3. Find the recurrence relation and solve for $a_n$
4. Apply any given initial conditions if applicable

Existence of Analytic Solution

The minimum radius of convergence is the distance to the nearest singular point

Singular points are where $P(x)$ or $Q(x)$ in $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$ don’t exist