For a physics problem I was told to set up a differential equation for the free fall in the gravitational field of the earth. The equation (via Newton) I've got is following:
$$\ddot{r} = - G M \frac 1 {r^2}$$
where $G$ is the gravitational constant, $M$ is the mass of the earth and $r$ is the distance to earths center of gravity. I was told to use this equation to find the velocity $\dot{r}$ but I have no idea how to do that, and it seems almost impossible to solve this differential equation analytically at all?
After your edit of the question
Hint: Try to multiply with $\dot r$ and integrate, $$ \int \ddot r\dot r\,dt=-GM\int \frac{\dot r}{r^2}\,dt. $$