A simple differential equation question.

[MathematicalPhysicist]

problem statement
A satellite travels in interstellar space, in its motion it collects stardust and its mass increases by $$\frac{dM}{dt}=\rho Av$$
A is the surface area of the satellite, rho is the stardust density, and v is the the speed of the satellite, at time t=0 the satelite speed is v0,and mass M0, you may assume that no external forces are acting upon the system, find the speed as function of time.

attempt at solution
well it looks simple $$0=\frac{dp}{dt}=\frac{dM}{dt}v+\frac{dv}{dt}M(t)$$
where $$M(t)=\rho Ax+M0$$ where x is the displacement the satelite goes, which yields: $$0=\rho A(\frac{dx}{dt})^2+\frac{d^2x}{dt^2}M(t)$$ here I am kind of stuck, if i ofcourse got it correct with the equation.
i think i can solve it with lowering the order of the equation, but first i want to see if i got the physics correct, so have i?

thanks in advance.

 

[muppet]

Looks ok to me! :)