I just learnt how to apply the power series method in differential equations and I'm now trying to understand the extended method of Frobenius. As an example, the textbook gives me Bessel's equation stating that I cannot proceed with the power series method. The equation is the following:
$$x^2y''+xy'+(x^2-v^2)y=0$$
Bringing this in the form we want for the Frobenius method we see that the coefficients of y' and y are analytic and we can use Frobenius:
$$y''+\frac{y'}{x}+(x^2-v^2)\frac{y}{x^2}=0$$
Why can't I just solve the equation using the power series method? I've already solved the following special Legendre equation using the power series method without any problem : $$(1-x^2)y'' -2xy' +2y=0$$
What is different in this equation from Bessel's equation that allows me to solve it?
Because Bessel's equation has a singularity at $x=0$, whereas the power series method assumes that the solution is analytic at $x=0$. In terms of the structure of the equation, you anticipate this singularity because the leading coefficient vanishes at $x=0$. Legendre's equation has no such singularity.