Consider the following non-homogeneous parabolic partial differential equation (PDE)
\begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + \epsilon \frac{\partial^{2}}{\partial \psi^{2}} \right)u(r, \psi) = -1, \end{align}
where $u(r, \psi): [0,1]\times[0,2\pi] \to \mathbb{R}^+$ and $\epsilon,\gamma \in \mathbb{R}^+$ are constant parameters. The boundary conditions are Dirichlet $u(r, \psi)|_{r=1} = 0$ and periodic $u(r, \psi+2\pi) = u(r, \psi)$.
How to find a particular solution of the above non-homogeneous PDE?
Thank you for your time and consideration.