# Rocket Equation with varying gravity

So, the rocket equation is

F_ext = m(dv/dt) + u(dm/dt)

where m is the mass of the rocket, v the velocity, u the effective exhaust gases speed, and F_ext the external forces on the system.

If we take a constant mass ejection rate p, and take the external force to be the gravitational attraction of a mass M, we recover the differential equation

(d^2x/dt^2) + GMx^(-2) = up/(m0 - pt)

where m0 is the initial mass (time t=0).

Can this type of differential equation be solved analytically? If so, how would one go about it?

Thanks,

Rudipoo