I have the following system of 2nd order PDEs in cylindrical coordinates,
$\frac{1}{r} \frac{\partial }{\partial \theta} \left( \frac{h}{3} a - \frac{\ell^2}{2} \nabla^2 a \right) + \frac{\partial }{\partial r} \left( \frac{h}{3} b - \frac{\ell^2}{2} \nabla^2 b \right)=0$
$\frac{\partial}{\partial \theta} \left[ \frac{1}{\mu} \left(\frac{h}{3} b - \frac{\ell^2}{2} \nabla^2 b \right) + b \right] - \frac{\partial}{\partial r} \left[ r \left(\frac{1}{\mu} \left( \frac{h}{3} a - \frac{\ell^2}{2} \nabla^2 a \right) + a \right) \right] = 0$
where h, $\mu$ and $\ell^2$ are constants, and $a$ and $b$ are the unknowns. These have a solution of the type, $a=r^{3/2} g_a (\theta)$ $b=r^{3/2} g_b (\theta)$
And I would like to uncouple the system to reduce it and solve it to obtain $g_a (\theta)$ and $g_b (\theta)$. Following the suggestions by @Yuriy S I reach a system like, $\frac{h}{3} r^{1/2} g_a' (\theta) - \frac{\ell^2}{2} r^{-3/2} \left(\frac{3}{4} g_a' (\theta) + g_a'''(\theta) \right) +\frac{h}{2} r^{1/2} g_b (\theta) + \frac{\ell^2}{4}r^{-3/2} \left(\frac{3}{4} g_b (\theta) + g_b'' (\theta) \right)=0$
$ \frac{1}{\mu} \left(\frac{h}{3} r^{3/2} g_b' (\theta) - \frac{\ell^2}{2} r^{-1/2} \left(\frac{3}{4} g_b' (\theta) + g_b''' (\theta) \right) \right) + r^{3/2} g_b' (\theta) - \frac{5}{2} r^{3/2} g_a (\theta) \left( 1 + \frac{h}{3 \mu} \right) - \frac{\ell^2}{4 \mu} r^{-1/2} \left( \frac{3}{4} g_a (\theta) + g_a'' (\theta) \right) = 0$
But I'm unable to reduce/solve it.
Thank you very much
Hint.
Assuming:
$$\nabla^2=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}$$
You have a system of 3rd order PDEs (because there are operators of the type $\frac{\partial}{\partial r} \nabla^2$ and $\frac{\partial}{\partial \theta} \nabla^2$).
You need to take the suggested forms for $a,b$:
$$a=r^{3/2} g_a (\theta) \qquad b=r^{3/2} g_b (\theta)$$
And substitute them in the system of equations. You will obtain a system of ordinary differential equations for $g_a,g_b$ in $\theta$.
This is a tedious process, but simple, so I leave it to you.
If the ansatz is correct, everything with $r$ should disappear from the equations (otherwise $g_a,g_b$ would depend on $r$ as well which is contrary to their definition).
You will obtain a system of 3rd order ODEs which in some cases can be decoupled (possibly with increasing the order of equations) or it can't be, in which case it should be solved numerically with usual methods.