Solving ordinary second-order differential equation
Solution 1:
$$\ddot{z} + \big( 3 \frac{\dot{x}}{x} + \frac{\dot{y}}{y} \big) \dot{z} =0$$
$$\dfrac {\ddot{z}}{\dot z} + 3 \frac{\dot{x}}{x} + \frac{\dot{y}}{y} =0$$ $$\ln \dot z+3\ln x+\ln y =C$$ $$\dot zx^3y=c_1$$ You can't go further and integrate this DE. You need to knwo what $x(t)$ and $y(t)$ are. You need further informations.