The PDE $$x\frac{\partial u}{\partial x}=y\frac{\partial u}{\partial y}$$ is separable, so we look for solutions of the form $u(x,t) = X(x)Y(y)$
The PDE can be rewritten using this solution as
= = $-\lambda$

Note: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\prime$. Do NOT use $X^\prime(x)$

Since these differential equations are independent of each other, they can be separated
DE in X: $= 0$
DE in T: $= 0$

These are both separable ODE's. The DE in X we separate as
$= \frac{-\lambda}{x}$
Integrate both sides, the constant of integration c going on the right side:
=
Using the funny constant algebra that $e^c = c$, solving for X we get X =
Since the differential equation in Y is the same we get Y = .
Finally u = .