Memoir on Integration of Ordinary Differential [1.2ex] Equations by Quadrature

TL;DR

This paper identifies a broad class of linear ordinary differential equations, including constant coefficient and Euler's equations, that can be integrated by quadrature, expanding the methods available for solving such equations.

Contribution

It introduces a new class of linear ODEs reducible to algebraic equations, providing a general method for their solution and unifying several known cases.

Findings

Identified a wide class of linear ODEs reducible to algebraic equations

Provided a method for solving these equations by quadrature

Included constant coefficient and Euler's equations as special cases

Abstract

The Riccati equations reducible to first-order linear equations by an appropriate change the dependent variable are singled out. All these equations are integrable by quadrature. A wide class of linear ordinary differential equations reducible to algebraic equations is found. It depends on two arbitrary functions. The method for solving all these equations is given. The new class contains the constant coefficient equations and Euler's equations as particular cases.