Integration of an arbitrary linear ODE
Peter C. Gibson
TL;DR
The paper develops a general, constructive framework to obtain explicit scalar solutions for homogeneous linear ODEs of order $n$ with variable coefficients by introducing the $n$-ary multex operator and associated trig operators. It constructs lower-order auxiliary functions $\varphi_1,\dots,\varphi_n$ whose interplay, via a Toeplitz-derived scheme, yields $n$ scalar solutions $\psi_{n,k}$ that form a local fundamental system through $\psi_{n,k}=T_{\varphi_{\eta(1)},\dots,\varphi_{\eta(n)};\eta^{-1}(n)}$ with $\eta=\pi_n^{k-1}$; the general local solution is $y=\sum_{k=1}^n y^{(k-1)}(0)\psi_{n,k}$. The method is demonstrated via Schrödinger impedance form and Orr-Sommerfeld equations, illustrating explicit, quadrature-based series expressions that avoid the traditional Dyson matrix approach. The work yields both a theoretical advance in explicit solvability and practical tools for applications in physics and stability analysis, with clear procedures for higher-order cases and explicit second-to-fourth order corollaries. Overall, it fills a long-standing gap in elementary ODE theory by providing a scalar, constructive solution framework for variable-coefficient linear ODEs.
Abstract
The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses this shortcoming in the basic theory. We introduce the multex integral operator, generalizing to several input functions the standard exponential primitive operator that is inverse to the logarithmic derivative. The multex operator serves to integrate in explicit form an arbitrary linear ordinary differential equation.