Is there an analytical solution to this differential equation? I am not very knowledgeable in the field so I apologize in advance if this question might look naive.
But is there an analytical solution to a differential equation that looks like this : 

$$\frac {\mathrm{d}y}{\mathrm{d}t} = a\cdot v(t) - \mathrm{b}\cdot y $$

My issue here is the function $v(t)$ and how I should deal with it.

Thanks
Yes,there is:

$$y(t)=ce^{-bt}+ae^{-bt}\int_1^t e^{bx} \nu(x)\,dx$$

where $c$ is a constant.
If I understand correctly, your equation is of the from
$$ \frac{\mathrm{d}y}{\mathrm{d}y} = -by(x) + h(x) $$

A solution then is available (http://eqworld.ipmnet.ru/en/solutions/ode/ode0103.pdf, just setting $g(x) = 1, f_1(x) = -b , f_0(x)=h(x) $)

$$ y = Ce^{-bx} + e^{-bx} \int e^{-bx} h(x)\mathrm{d}x $$