I am trying to solve the particle in a disk problem and arrived at the radial part ODE, shown in bellow.
$$r^2\frac{\mathrm{d^2} \phi_r}{\mathrm{d^2} r} + r\frac{\mathrm{d} \phi_r}{\mathrm{d} r} + (\omega^2r^2-\ell^2)\phi_r = 0$$
What I found, after some research, is that this ODE is a special case of the Bessel ODE, or, as shown in here a scaled version of that ODE.
The problem I am facing with the solving of this particular ODE is that in the end of the series expansion I arrive at the following equation.
$$\displaystyle \left.\sum_{n=2}^\infty\middle[\omega^2a_{n-2} - (\ell^2 - n^2)a_n \right] r^n = \ell^2a_0 + (\ell^2-1)a_1r$$
This, in turn lead me to the conclusions that $a_0=a_1=0$ and $\displaystyle a_n = \left(\frac{\omega^2}{\ell^2 - n^2}\right) a_{n-2}$ , followed by the conclusion that $a_n=0\ \forall\ n$
Am I doing something wrong? What it is?
A more detailed devolopment of this solution can be found here.