\displaystyle \dot{z} = \frac{x y}{z} \ \Rightarrow \ \dot{z} = \frac{A B e^{-t}}{z} \ \Rightarrow \ z \, dz = A B e^{-t} \, dt \ \Rightarrow \ \frac{1}{2} z^2 = -A B e^{-t} + C