Integrating Factors and First Integrals for a Class of Higher Order Differential Equations
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 $Ω\subset\R^{n+1}$. 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.