An explicit Maclaurin series solution to non-autonomous and non-homogeneous evolution equation, Omega Calculus and associated applications.

This article presents a novel integral-free representation of the solution to a non-autonomous, non-homogeneous evolution equation using the Omega Calculus (OC), also known as MacMahon's Partition Analysis. The approach generalizes recent results by providing explicit, non-recursive formulas for the Maclaurin series coefficients of the solution derived from the Peano–Baker series, and addresses the inverse problem of determining the generator of dynamics when the solution is known and analytic in a Hilbert space setting. Applications include solving recursion relations arising in vibration problems of non-uniform Euler–Bernoulli beams, where the method yields explicit series solutions. The work demonstrates the versatility of OC beyond its combinatorial origins, extending its utility to operator-valued differential equations and applied mathematics contexts.