It is usually straightforward to derive the recurrence relation required to generate the power sereis solution to the differential equation. The method does not always produce a solution however, and we need to know when it fails.
Theorem (The Convergence Theorem)
If
where
and
are polynomials and all the zeros of
are real, any power series solution
about
has an interval of convergence
where
is the minimum distance from
to a zero of
in factis a lower bound for the radius of convergence, but the radius of convergence is equal toexcept in very special circumstances.
We might expect no convergence for a series expansion about
whereis a zero of Consider the differential equation
For this equation
which is zero at
If we assume a power series solution of the form
then
Substitution into the original differential equation gives
which after re - indexing of the first summation term becomes
or
Hence
and
Hence
and the series mothod gives
The separation of variables method gives
but the series method failed to detect this solution because
is a zero of