1. In a particular physics model we are interested in finding $\phi(x, y)$ governed
by
$\frac{\partial^2 \phi}{\partial x^2} - \frac{9}{b} \frac{\partial \phi}{\partial x} + \frac{b}{x} \frac{\partial^2 \phi}{\partial y^2} = 0$
where $b$ is a real constant. The domain of $\phi(x, y)$ is $-b < x < b$ and $y \ge 0$.
The boundary conditions are
$\phi(0, y) = 0$
$\phi(b, 0) = \lambda$
$\phi(x, 0) = \text{4th order polynomial}$
where $\lambda$ is a constant. The solution $\phi$ also decreases with increasing $y$.
Find the complete solution $\phi(x, y)$.
