Partial differential equation/ wave equation

Question

I literally have no idea how to resolve this equation:

ψ is a real (and differentiable) function. Show that this functions satisfies the expression. $$f(x,y)=x^2ψ(3x+y^2)\\ 2xy(∂f/∂x)-3x(∂f/∂y)=4yz$$

Answer 1

Hint: Let $u=3x+y^2$ then with $f(x,y)=x^2\psi(u)$ we write $$\dfrac{\partial f}{\partial x}=2x\psi(u)+x^2\dfrac{d\psi}{du}(3)$$ $$\dfrac{\partial f}{\partial y}=x^2\psi(u)2y$$ With substitution you find $2xy\dfrac{\partial f}{\partial x}-3x\dfrac{\partial f}{\partial y}=4yf$.