Derivatives of r with respect to time may be rewritten as derivatives of u with respect to angle.
F=m(\ddot{r}-r\dot{\theta }^2)=-m\left(h^{2}u^{2}\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+h^{2}u^{3}\right)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u\right) .