In mathematics, the power series method is used to seek a power series solution to certain differential equations. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

Method

Consider the second-order linear differential equation

Suppose a₂ is nonzero for all x. Then we can divide throughout to obtain

Suppose further that a₁/a₂ and a₀/a₂ are analytic functions. In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

The power series method tells us we may be able to construct a power series solution

If a₂ is zero for some x, then the Frobenius method, a variation on this method, is suited to deal with so called singular points. In mathematics, the Frobenius method describes a way to find an infinite series solution for a second-order ordinary differential equation of the form We can divide through by z2 to obtain a differential equation of the form which we can solve with regular power series methods if p(z...

Example usage

Let us look at the Hermite differential equation,

We can try and construct a series solution

Substituting these in the differential equation

Making a shift on the first sum

Now, if this series is a solution, all these coefficients must be zero, so:

We can rearrange this to get a recurrence relation for A_k+2.

Now, we have

We can determine A₀ and A₁ if there are initial conditions, ie., if we have an initial value problem.

So, we have

and the series solution is

which we can break up into the sum of two linearly independent series solutions:

which can be further simplified by the use of hypergeometric series (which goes beyond the scope of this article). In mathematics, a hypergeometric series is the sum of a sequence of terms in which the ratios of successive coefficients k is a rational function of k. ...