Problem #2:
Use separation of variables with $\lambda = \frac{1}{16}$ to find a
product solution to the following partial differential
equation,
$y \frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial y} = 0$
that also satisfies the conditions $u(0, 0) = 9$ and
$u_x(0, 0) = 7.$