Differential equation with arbitrary function $f(t)$.

Question

I need help with this differential equation:

$$\frac{dv}{dt} + k*v = \frac{f(t)}{m}.$$

I need to solve for $v$. $K$ and $M$ are constants and $f(t)$ can be any arbitrary function. $V$ is velocity which also depends on $t$. I know it should be through integration factor but the $f(t)$ is confusing me.

Answer 1

Multiplying both sides of the equation by $e^{k t}$ it is noticed that $$\frac{d}{dt} \, (e^{k t} \, v) = \frac{e^{k t} \, f(t)}{m}.$$ At this point integration can be applied to obtain $$v(t) = \frac{1}{m} \, \int^{t} e^{k (u-t)} \, f(u) \, du + c_{0}$$