Use separation of variables to solve the partial differential equation $x^2 \dfrac{\partial^2u}{dx^2}+\dfrac{\partial^2u}{dy^2}=0$

Question

After using separation of variables on the following equation

$$x^2 \dfrac{\partial^2u}{dx^2}+\dfrac{\partial^2u}{dy^2}=0$$

I got to the following equation

$$x^2 X'' + a^2X = 0$$

How do I solve it? Is it non-linear?