A Substitution Based Method for Solving Non-Homogeneous Linear Differential Equations

We develop a unified, substitution-based framework for solving non-homogeneous linear ordinary differential equations (ODEs) via the systematic factorization of the associated differential operator. By decomposing higher-order operators into a nested sequence of first-order linear factors, the non-homogeneous problem is resolved through successive integrations using integrating factors. This construction yields explicit integral representations of particular solutions, providing a direct derivation of the convolution kernel and Green’s function without requiring the method of undetermined coefficients, variation of parameters, or Laplace transforms. For second-order equations, we clarify the structural origins of resonance and oscillatory behavior and extend the approach to variable-coefficient settings, specifically Cauchy–Euler equations.