When does the Power Series Method for Solving 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)
Ifwhereandare polynomials and all the zeros ofare real, any power series solutionabouthas an interval of convergencewhereis the minimum distance fromto a zero ofin 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 aboutwhereis a zero of Consider the differential equationFor this equationwhich is zero at
If we assume a power series solution of the formthen
Substitution into the original differential equation giveswhich after re - indexing of the first summation term becomesor
Henceand the series mothod gives
The separation of variables method givesbut the series method failed to detect this solution becauseis a zero of