A body of mass m is projected vertically upwards with speed u. Air resistance is equal kv^2. Find the speed when next at the point of projection.
$\displaystyle \frac{dv}{dt} = - g - \frac{k}{m} v^2$
$\displaystyle \Rightarrow \frac{dt}{dv} = - \frac{1}{g + \frac{k}{m} v^2} = - \frac{m}{mg + k v^2}$ subject to the boundary condition that v = u at t = 0.
Solve this differential for t as a function of v and then make v the subject. Get x from v and use it to find x when v = 0 (the maximum height).
Now set up and solve the differential equation for the downwards motion. Find v when x = distance found above.