If I'm given the homogenous first-order linear ODE in one dimension
$\frac{dx}{dt} = ax(t), \ \ x_0 = 1$,
a straightforward calculation shows that
$x(t) = e^{a t}$.
Now if I add some driving function $f$ that depends only on the current time $t$, the dynamics becomes
$\frac{dx}{dt} = ax(t) + f(t), \ \ x_0 = 1$.
The solution to the inhomogenous ODE is
$x(t) = e^{a t} + \int_0^t e^{a (t-u)}f(u)du$,
so we can relate the solution of the inhomogenous equation to that of the homogenous equation and a certain operation on the driving function (in this case, a kind of exponential damping or amplification).
I'm wondering if there are any results that generalise this to the nonlinear setting (still first-order). If I know the solution $x(t)$ of the homogenous equation, is there some result on the relationship between $x$ and the solution to the ODE once I include a driving function?
If no general result exists, are there conditions that I can impose on the dynamics to deal with this kind of thing in special cases?