Mechanics question about changing mass differential equation.

I'm stuck on this question and was wondering if anyone could help me out.

What I did:

Okay so I set up to times in which to analyse the rocket. t=0 and t=T. From this we should be able to apply impulse momentum principle to get a differential equation that we can solve to find x(t) then presumably taking the limit as t ----> infinity will yield the correct expression.

At t=0 I said the total mass of the system was mass of the rocket + mass of the fuel.

M_r+M_f=M_total. (1)

Also we know that d(M_f)/dt=-k and we can solve this to find the mass of the remaining fuel on the rocket is the mass of the initial fuel on the rocket-kt.

So I want to say Impluse=change in momentum and the use this to get out the differential equation.

My problem is somewhat getting this information into the correct variables that allow me to work out exactly what I want to do without getting in a big mess which is what is happening at the moment.

Also I don't know how to determine the impulse (I know it's f*t=m*a*dt) but which masses do I use and how does the negative direction of the fuel play a part in this expression?

Really need help here so anything is appreciated.