Solving a second order PDE already in canonical form

Question

I am trying to solve the second order PDE, $$\partial_x \partial_y u + \frac{1}{x} \partial_y u - y^2 = 0.$$ My problem is this is in canonical form and I don't see how to apply the method of characteristics, since $\frac{\mathrm dy}{\mathrm dx} = \frac{b^2 + \sqrt{b^2-4ac}}{2a}$ where $a$ is the function in front of $\partial_{xx}u$ but there is no such term here so $a=0$.

I am not asking for a solution, I would just like to know what method I can apply here to obtain the general solution, which I am imagine will be an arbitrary function of some function of $x$ and $y$. Am I suppose to perform an inverse transformation out of canonical form?

Answer 1

Let $z(x,y)=u_y(x,y)$. Then you get a first-order PDE for $z$. (Actually an ODE, for each fixed $y$.)