Let $A:\mathbb {R} \to M_n (\mathbb{R}) $ and $h: \mathbb {R} \to \mathbb {R}^n$ be smooth bounded functions. Consider $\Phi : \mathbb {R} \to M _n (\mathbb {R})$ as a fundamental matrix of the ODE $$\dot {y} = A (t) y. $$
Suppose that we know that $\varphi: \mathbb{R}\rightarrow \mathbb{R}^n$ solve the following ODE
$$\dot{y} = A(t)y + h(t), $$
and $\varphi(t) \to 0,$ when $t \rightarrow -\infty $.
Note that using the constants variantion formula
$\varphi (t) = \Phi (t)\left [\Phi^{-1}(t_0) \varphi (t_0) +\int_{t_0}^{t} \Phi^{-1} (s) h (s) ds\right],$ forall $t_0$ $\in $ $\mathbb {R} $.
I would like to know if under these assumptions
$$\varphi (t) = \Phi (t)\int_{-\infty}^{t} \Phi^{-1} (s) h (s) ds ?$$
Can anyone help me?