Existence and uniqueness of solution for a seemingly trivial 1D non-autonomous ODE So I was trying to do some existence and uniqueness results beyond the trivial setting. So consider the 1D non-autonomous ODE given by
$\dot{y} = f(t) - g(t) y $ where $f,g \geq 0$ are integrable and $f(t),g(t) \rightarrow 0$ for $t \rightarrow \infty$. How would I go about proving the existence and uniqueness for the solution of such an ODE for $t \rightarrow \infty$?
Just by a contraction argument?

Assuming you meant $\dot{y} = f(t) - g(t) y$, this is a linear differential equation and has the explicit solutions $y(t) = \mu(t)^{-1}  \int \mu(t) f(t)\ dt$ where 
$\mu(t) = \exp(\int g(t)\ dt)$, from which it is clear that you have global existence of
solutions.  Uniqueness follows from the standard existence and uniqueness for ODE. 
