$$ \frac{\partial^2{u(x,y)}}{\partial y^2} -x^2 {u(x,y)} = 0$$
If i have something like this^. is it okay to use the standard technique for solving second order homogeneous differential equations?
i.e. $ r^2 = x^2 $
Which leads to $${u}(x,y) = A e^{x y} + B e^{-x y}$$
Just wondering because u is in terms of x but the differentials are with respect to y.
What if
$$u(x,y)=e^{xy}?$$
The first partial wrt $y$ is
$$xe^{xy}$$
and the second one is then
$$x^2e^{xy}=x^2u(x,y).$$