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

We present a substitution-based framework for solving non-homogeneous linear ordinary differential equations by factorizing the associated differential operator into first-order factors. This reduces the original higher-order equation to a nested sequence of first-order linear equations, each solvable by an integrating factor. The method gives explicit integral representations of particular solutions and directly constructs the corresponding causal Green’s functions. For constant-coefficient second-order equations, the method recovers the standard kernels for distinct real roots, repeated roots, and complex conjugate roots. The approach also applies to variable-coefficient equations whenever a first-order factorization is available, including Cauchy-Euler equations. Finally, we show that the factorization problem for second-order variable-coefficient equations is equivalent to finding a particular solution of an associated Riccati equation.