I would like to solve the following differential equation:
$$y''(x)+\left[a^2-\frac{b(b+1)}{x^2}\right]y(x)=0$$
where $a>0$ and $b=0,1,2,...$
I tried to solve this equation by applying a Laplace transform to both sides but this yielded another differential equation that I was unable to solve:
$$(s^2+a^2)Y''(s)+4sY'(s)+[2-b(b+1)]Y(s)=0$$
where $Y(s)=\mathscr{L}_x[y(x)](s)$. Does anybody know how to solve either the original or the transformed equation?
By introducing $f(x)$ which is related to $y(x)$ via $$y(x)= x^{1/2} f(x)$$ your ODE is transformed into $$x^2 f''(x) + x f'(x) + \left[a^2 x^2 - \left(b + \frac12\right)^2\right] f(x)=0.$$
This is Bessel's differential equation with the general solution $$ f(x) = c_1 J_{b+1/2}(a x) + c_2 Y_{b+1/2}(a x).$$