Ordinary Points and Power Series Solutions

Power series methods provide systematic techniques for solving linear ODEs near ordinary and singular points, yielding solutions as convergent infinite series.

Analytic ODEs and Ordinary Points

Definition9.1Ordinary and singular points

Consider the second-order linear ODE $y'' + P(x)y' + Q(x)y = 0$ . A point $x_0$ is an ordinary point if both $P(x)$ and $Q(x)$ are analytic (have convergent power series expansions) at $x_0$ . Otherwise, $x_0$ is a singular point.

Equivalently, for the standard form $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ with analytic coefficients, $x_0$ is ordinary if $a_2(x_0) \neq 0$ , and singular if $a_2(x_0) = 0$ .

Theorem9.1Existence of power series solutions at ordinary points

If $x_0$ is an ordinary point of $y'' + P(x)y' + Q(x)y = 0$ , then there exist two linearly independent solutions of the form:

$y(x) = \sum_{n=0}^{\infty} a_n (x - x_0)^n,$

and the radius of convergence of each series is at least the distance from $x_0$ to the nearest singular point (in $\mathbb{C}$ ).

The Recurrence Method

ExampleAiry equation

The Airy equation $y'' - xy = 0$ has $x_0 = 0$ as an ordinary point (no finite singular points, so $R = \infty$ ). Substituting $y = \sum a_n x^n$ :

$\sum_{n=2}^{\infty} n(n-1)a_n x^{n-2} - \sum_{n=0}^{\infty} a_n x^{n+1} = 0.$

Re-indexing: $\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n = \sum_{n=1}^{\infty}a_{n-1}x^n$ . So $2a_2 = 0$ (hence $a_2 = 0$ ) and for $n \geq 1$ :

$a_{n+2} = \frac{a_{n-1}}{(n+2)(n+1)}.$

With $a_0$ and $a_1$ free, the two linearly independent solutions are the Airy functions $\mathrm{Ai}(x)$ and $\mathrm{Bi}(x)$ , both entire functions.

ExampleHermite equation

$y'' - 2xy' + 2ny = 0$ ( $n$ a parameter). At $x_0 = 0$ (ordinary point), substitute $y = \sum a_k x^k$ :

$a_{k+2} = \frac{2(k - n)}{(k+2)(k+1)} a_k.$

When $n$ is a non-negative integer, the series terminates, yielding the Hermite polynomials $H_n(x)$ . For $n = 0$ : $H_0 = 1$ . For $n = 1$ : $H_1 = 2x$ . For $n = 2$ : $H_2 = 4x^2 - 2$ .

Convergence Analysis

RemarkDetermining the radius of convergence

The radius of convergence $R$ is at least $\min(R_P, R_Q)$ , where $R_P$ and $R_Q$ are the radii of convergence of $P(x)$ and $Q(x)$ about $x_0$ . This is equivalent to the distance from $x_0$ to the nearest singularity of $P$ or $Q$ in the complex plane.

For example, $(1+x^2)y'' + y = 0$ has $P = 0$ , $Q = 1/(1+x^2)$ , which is singular at $x = \pm i$ . Series about $x_0 = 0$ converge for $|x| < 1$ .

This illustrates a key insight: complex singularities limit the radius of convergence of real power series solutions.

RemarkPractical computation

To find the first $N$ terms of a power series solution:

Write $y = \sum_{n=0}^{N} a_n(x-x_0)^n$ .
Substitute into the ODE and collect powers.
Set each coefficient to zero to obtain a recurrence relation for $\{a_n\}$ .
Express $a_n$ in terms of $a_0$ (and $a_1$ for second-order).
The two free constants $a_0$ , $a_1$ correspond to the two linearly independent solutions.