Bessel equation

Question

The Bessel equation is: $$x^2\frac{\mathrm d^2y}{\mathrm dx^2}+x\frac{\mathrm dy}{\mathrm dx}+\left(x^2-m^2\right)y=0$$

Does the fact that the series is convergent assist in the solving of the question? Initially I am thinking this is just simple differentiation and substiution - am I right in this assertion or is there more to it.

Answer 1

The solution should indeed be a straightforward substitution and term-by-term differentiation. The assumption that the series is convergent is implicitly used in two ways:

1. to give you an a priori well-defined function to substitute;
2. to allow term-by-term differentiation, by the fact that a convergent power series can be differentiated term-by-term within its radius of convergence.

Other than these points, I don't think the convergence issue will affect your solution.