A generalization of d’Alembert formula

In this paper we find a closed form of the solution for the factored inhomogeneous linear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \prod\limits_{j = 1}^n {\left( {\frac{d} {{dt}} - A_j } \right)} u(t) = f(t). $$\end{document} Under the hypothesis A1, A2, …, An are infinitesimal generators of mutually commuting strongly continuous semigroups of bounded linear operators on a Banach space X. Here we do not assume that Ajs are distinct and we offer the computational method to get explicit solutions of certain partial differential equations.