Question

Solve the PDE ∂z/∂x + ∂z/∂y = z²

Answer 1

$\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}=z^2\\ \text{The auxillairy equation are, }\\ dx=dy=\frac{dz}{z^2}\\ \text{Integrate the first two fractions. We have that; }\\ x=y+a\\ x-y=a\\ \text{Integrate the last two fractions }\\ y=-\frac{1}{z}+b\\ y+\frac{1}{z}=b.\\ \text{Hence the solution is } f(a,b)=f(x-y,y+\frac{1}{z})=0.$