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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?

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    $\begingroup$ Let $X(x)=x^r$ and solve for two values of $r$, leading to two independent particular solutions. $\endgroup$ – JJacquelin Oct 9 '16 at 10:34

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