Power series solution of differential equations

In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

Consider the second-order linear differential equation

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

Suppose further that a₁/a₂ and a₀/a₂ are analytic functions.

The power series method calls for the construction of a power series solution

If a₂ is zero for some z, then the Frobenius method, a variation on this method, is suited to deal with so called singular points. The method works analogously for higher order equations as well as for systems.

Let us look at the Hermite differential equation,

We can try to construct a series solution

Substituting these in the differential equation

Making a shift on the first sum

If this series is a solution, then all these coefficients must be zero, so for both k=0 and k>0:

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, i.e. 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.

A much simpler way of solving this equation (and power series solution in general) using the Taylor series form of the expansion. Here we assume the answer is of the form

If we do this, the general rule for obtaining the recurrence relationship for the coefficients is

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