homework and exercises - Analytical solution of Liouville's equation for classic harmonic oscillator

I'm interested in the analytical solution of the simple PDE:

$$\frac{\partial f}{\partial t} - m\omega^2x\frac{\partial f}{\partial p}+ \frac{p}{m} \frac{\partial f}{\partial x} ~=~ 0.\tag{1}$$

With: $$f(x,p;t\!=\!0)~=~f_0(x,p) \quad \mbox{arbitrary smooth},\tag{2}$$ $$x(t)~=~x_0 \cos(\omega t) + \frac{p_0}{m\omega}\sin(\omega t),\tag{3}$$ $$p(t)~=~p_0 \cos(\omega t) - m\omega x_0 \sin(\omega t).\tag{4}$$

And $x_0, p_0$ constants.