Partial derivatives of multivariable functions

Question

I'm having a lot of trouble with this question and I can't seem to even figure out where to start. Would anyone be able to help out with this? Thanks :)

Answer 1

$$\frac{\partial G}{\partial t}=A\frac{\partial F}{\partial z}$$ $$\frac{\partial^2 G}{\partial \gamma^2}=\frac{\partial}{\partial \gamma} \left(\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \right)%(\gamma+s,\gamma-s,At) =\frac{\partial^2 F}{\partial x^2}+2\frac{\partial^2 F}{\partial y \partial x}+\frac{\partial^2 F}{\partial y^2} $$ $$\frac{\partial^2 G}{\partial s^2}=\frac{\partial}{\partial s} \left(\frac{\partial F}{\partial x} - \frac{\partial F}{\partial y} \right)%(\gamma+s,\gamma-s,At) =\frac{\partial^2 F}{\partial x^2}-2\frac{\partial^2 F}{\partial y \partial x}+\frac{\partial^2 F}{\partial y^2} $$ Thus, $$\frac{\partial^2 G}{\partial s^2}+\frac{\partial^2 G}{\partial \gamma^2}=2\frac{\partial F}{\partial z} $$ and we got that $A=2$