University Maths Notes: Calculus – When Does the Power Series Solutions Method for Differential Equations Break Down?
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 to
except
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