I am interested in a simple parametric function: $$y(t) = \frac{a}{b} (1-e^{-b t})$$
Where $a$ and $b$ are real parameters greater than zero; $y(0)=0$; and $t$ is time in discrete units, e.g. months or years.
If the value of $a$ can vary arbitrarily by time unit, $a(t)$, is there a solution for $y(t)$?
Note that this function is the solution to the ordinary differential equation: $$ \frac{dy}{dt} = a- b y $$
I am a non-mathematician noob, so any input would be appreciated. Many thanks in advance
By laplace transform: $$ \mathcal{L}\{y'+by(t)\}=\mathcal{L}\{a(t)\} $$ $$ sY(s)-y(0)+bY(s) = A(s) $$ $$ Y(s)(s+b)=A(s)+y(0) $$ $$ y(t) = \mathcal{L}^{-1}{{A(s)\over s+b}}+y(0)e^{-bt} $$ Which can be written as:$$ \int_0^ta(u)e^{-b(t-u)}du$$ This is the general formula