I am trying to solve the velocity as a function of the time: $$ ma=F_\text{Thrust}-F_\text{Drag}\Leftrightarrow m \frac {\rm d}{{\rm d}t} v(t) = \frac p {v(t)} - k\cdot(v(t))^2 $$
where $m$ is the mass [kg], $k$ is the drag coefficient and $p$ is the power [W]
I wonder if it is even possible to solve it analytical? If possible, how do I approach it?
Rewrite it as $m\dfrac{v}{p - kv^3} \dfrac {{\rm d}v}{{\rm d}t}= 1$, decompose $\dfrac{v}{p - kv^3}$ into partial fractions, and integrate.