Power Series Solutions of Differential Equations Cheat Sheet

Key Facts

• A power series solution about $x_0$ has the form $y=\sum_{n=0}^{\infty}a_n(x-x_0)^n$.
• Termwise differentiation is valid inside the radius of convergence, so $y'=\sum_{n=1}^{\infty}n a_n(x-x_0)^{n-1}$ and $y''=\sum_{n=2}^{\infty}n(n-1)a_n(x-x_0)^{n-2}$.
• An ordinary point for $P(x)y''+Q(x)y'+R(x)y=0$ satisfies $P(x_0)\neq 0$ and the functions $\frac{Q(x)}{P(x)}$ and $\frac{R(x)}{P(x)}$ are analytic at $x_0$.
• After substitution, rewrite every sum using the same power, usually $(x-x_0)^n$, then set each coefficient equal to $0$.
• A recurrence relation expresses coefficients such as $a_{n+2}$ in terms of earlier coefficients, often leaving $a_0$ and $a_1$ free for a second order equation.
• For a regular singular point, use the Frobenius form $y=\sum_{n=0}^{\infty}a_n(x-x_0)^{n+r}$ with $a_0\neq 0$.
• The indicial equation comes from the lowest power of $(x-x_0)$ after substituting the Frobenius series, and its roots determine possible values of $r$.
• The radius of convergence is limited by the distance from $x_0$ to the nearest singular point of the differential equation coefficients.

Vocabulary

Power series solution

A solution written as an infinite polynomial $y=\sum_{n=0}^{\infty}a_n(x-x_0)^n$ with coefficients chosen to satisfy a differential equation.

Ordinary point

A point $x_0$ where the differential equation can be divided into standard form and all coefficient functions are analytic.

Singular point

A point where the leading coefficient is $0$ or where the standard form coefficient functions fail to be analytic.

Regular singular point

A singular point $x_0$ where $(x-x_0)\frac{Q(x)}{P(x)}$ and $(x-x_0)^2\frac{R(x)}{P(x)}$ are analytic.

Recurrence relation

An equation that defines later series coefficients in terms of earlier coefficients, such as $a_{n+2}=\frac{a_n}{(n+2)(n+1)}$.

Indicial equation

The algebraic equation for the Frobenius exponent $r$ obtained from the lowest power term after substitution.