I have a PDE $$\frac{1}{r} \frac{\partial}{\partial r} \left ( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{1}{R_0^2} \frac{\partial^2 f}{\partial \zeta^2} = \frac{2r \cos \theta}{R_0^3} \frac{\partial^2 f}{\partial \zeta^2}.$$
At $\frac{r}{R_0} \to 0$ $RHS \to 0$ and I have a solution in the form $f_0 = \varphi_1(r) \sin(\theta - \zeta) + \varphi_2 (r) \sin (\theta+\zeta)$, where I can easily find a solution for $\varphi_i(r)$.
So next I want to substitute $f_0$ in the right-hand side to get the "first correction" $f_1$ and I get problems with finding $f_1$. Are there any other ways to find an approximate solution if $\frac{r}{R_0}$ is small?