a=dV/dt
Write the original equation as a differential equation.
V(mdV/dt + f)=P
Solve with the idea of separating variables, and get it by separating variables first.
VdV/(P-fV)=dt/m
that is
[P/(P-fV)- 1]dV=fdt/m
Both sides do definite integral together, corresponding to initial state and final state. So v integrates from 0 to v, and t integrates from 0 to t.
What I got was:
[P*ln(P/P-fV)]/f -V = ft/m.
The latter simplification is of little significance, because it is not so easy to write a table like this: V=f(t), so it should be possible to keep it like this. The reason is that the left side represents speed and the right side is just the form of speed, so it can be expressed like this.