Solving Linear Differential Equations by recursion and integrating factors

TL;DR

This work develops a recursive framework for solving ordinary differential equations and combines it with generalized integrating factors to tackle both first- and second-order linear ODEs encountered in physics. For first-order equations, the recursion reproduces the classical integrating-factor solution and can be cast as a Volterra equation, yielding the standard $y(x)=C_1 e^{-rac{}{} }+e^{-rac{}{} }igl(igr)$ form. For second-order equations, the method constructs two integrating factors $oldsymbol{eta}$ and $oldsymbol{ extalpha}$ by solving $oldsymbol{ extalpha}''-h(x)oldsymbol{ extalpha}=0$ with $h=-q+rac{1}{2}p'+rac{p^2}{4}$, enabling a reduction to a first-order problem and explicit expressions in terms of $oldsymbol{ extalpha}/oldsymbol{eta}$ and double integrals. The authors demonstrate the approach on constant-coefficient, Cauchy-Euler, Airy, Legendre, Hermite, and Chebyshev equations, recovering known homogeneous solutions such as $y_h=C_1 e^{rac{-b\

Abstract

In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in classical differential equations encountered in physics, inclusive of equations with variable coefficients, particularly when a pattern within the recursion is identifiable, thus enabling the derivation of an explicit expression for $y(x)$.