General Solution of Second Order Linear Ordinary Differential Equation

TL;DR

This paper introduces a direct formula for solving general second order linear ODEs, utilizing a Riccati equation to determine parameters, and covers special cases with variable and constant coefficients.

Contribution

It provides a novel explicit solution formula for second order linear ODEs based on a Riccati equation, extending existing methods.

Findings

Derived a direct solution formula for second order linear ODEs.

Established special case solutions for variable and constant coefficients.

Presented formulas when one solution is known.

Abstract

In this paper we present a direct formula for the solution of the general second order linear ordinary differential equation as our main result such that the parameters required for the formula are determined using another differential equation, which happens to be a Riccati Equation. We also derive other results based on the main result which include special cases for the concerned differential equation with variable coefficients, formula for solution of concerned differential equation with constant coefficients and formula for the solution of the concerned differential equation with one complimentary solution known.