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In this article we present a necessary and sufficient condition for the existence of $psi$-bounded solution on $mathbb{R}$ of the nonhomogeneous linear differential equation $x'=A(t)x+f(t)$. We associate that with the condition of the concept $psi$-dichotomy on $mathbb{R}$ of the homogeneous linear differential equation $x'=A(t)x$.