> 
> A rocket has velocity $v$. Burnt fuel of mass $\Delta m$ leaves at
>  velocity $v-7$. Total momentum is constant:
>  $$mv=(m-\Delta m)(v+\Delta v) + \Delta m(v-7).$$
>  What differential equation connects $m$ to $v$? Solve for $v(m)$ not $v(t)$, starting from $v\_0 = 20$ and $m\_0 = 4$.
> 
> 
> 

Simplifying the equation for momentum, we have $m\Delta v-\Delta m\Delta v=7\Delta m$. Dividing by $\Delta m$, we get $m\frac{\Delta v}{\Delta m}-\Delta v=7$. As $\Delta m\to0$, $m\frac{dv}{dm}=7$ or $\frac{dv}{dm}=\frac7m$. Then, $v=7\ln m+C$.

At this point, I am getting a feeling that something is not right, because this is a section about exponentials, and logarithms are in the next section. Even if I ignore this, I do not know how to proceed with $v\_0$ and $m\_0$, since that seems to involve $t=0$. Could you provide some tips on this problem?