This equation is true for a rocket:

where, m is mass, v is velocity, g is 9.81, k is constant, u is the velocity at which mass is being ejected and gamma is the rate at which mass is being ejected.

I have drawn up two methods, the Euler's method and perturbation theory. I need a third method and can't seem to think of any. All suggests would be great. 

Have a look at http://star-www.dur.ac.uk/~tt/MSc/Lecture2.pdf

It recommends Runge-Kutta for accuracy.

Edited February 14, 20232 yr by sethoflagos

On 2/14/2023 at 8:31 PM, sethoflagos said:

Have a look at http://star-www.dur.ac.uk/~tt/MSc/Lecture2.pdf

It recommends Runge-Kutta for accuracy.

Do you have any analytical method for this?

1 hour ago, Flemish said:

Do you have any analytical method for this?

It's many years since I used these methods. There are many hits for Runge-Kutta on the internet but I've no idea which would be best for you. I think the appropriate form to use is RK4. 

Maybe others can recommend an appropriate text book. It might help if you indicated your preferred language and level of maths education.