Integrating Factors and First Integrals for a Class of Higher Order Differential Equations

TL;DR

This paper extends the method of finding integrating factors to higher order differential equations, enabling their transformation into exact equations and reduction of order, with explicit forms and detailed analysis for third order cases.

Contribution

It introduces a generalized approach for deriving integrating factors for a broad class of higher order differential equations, including explicit forms and special cases.

Findings

Explicit forms of integrating factors derived for the class of higher order equations

Reduction of order achieved through transformation into exact equations

Detailed analysis and examples for third order differential equations

Abstract

If the $n-th$ order differential equation is not exact, under certain conditions, an integrating factor exists which transforms the differential equation into an exact one. Hence, its order can be reduced to the lower order. In this paper, the principle of finding an integrating factor of a none exact differential equations is extended to the class of $n$-th order differential equations \begin{align} F_n\left(t,y,y^\prime,y^{\prime\prime},\ldots,y^{(n-1)}\right)y^{(n)}&+F_{n-1}\left(t,y,y^\prime,y^{\prime\prime},\ldots,y^{(n-1)}\right)y^{(n-1)}+\cdots +\nonumber\\ &+F_{1}\left(t,y,y^\prime,y^{\prime\prime},\ldots,y^{(n-1)}\right)y^{\prime}+F_{0}\left(t,y,y^\prime,y^{\prime\prime}\ldots,y^{(n-1)}\right)\nonumber\\ &=0,\nonumber \end{align} where $F_0,F_1,F_2, \cdots,F_n$ are continuous functions with their first partial derivatives on some simply connected domain $\Omega \subset\R^{n+1}$. In particular, we prove some explicit forms of integrating factors for this class of differential equations. Moreover, as a special case of this class, we consider the class of third order differential equations in more details. We also present some illustrative examples.