So this is NOT "solving" a differential equation, it is creating a differential equation.

The derivative of a constant is 0 so start differentiating!

Differentiate with respect to x

\(\displaystyle 2z\frac{\partial z}{\partial x}= 3ax^2\)

Differentiate again with respect to y

\(\displaystyle 2z\frac{\partial^2z}{\partial x\partial y}+ 2\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}= 0\)