The inverse integrating factor for some classes of \(n\)-th order autonomous differential systems
Consider an \(n\)-order autonomous differential system \[ \dot{x}=p(x), \eqno{(1)} \] where \(x=(x_1,x_2,\ldots,x_n) \in D \subseteq \mathbb{R}^n\) and \(p(x)=(p_1(x),p_2(x),\ldots,p_n(x))\), with \(p_i\in C^1(D, \mathbb R)\) functions in \(D\). The vector field associated to system (1) is denoted by \[ \mathcal{X}= \sum_{i=1}^n p_i(x) \frac{\partial}{\partial x_i}, \] and its divergence by \(\mathrm{div}\mathcal{X}= \sum_{i=1}^n \frac{\partial p_i}{\partial x_i}\). A non-constant function \(u\in C^1(D,\mathbb R)\) is called an \textit{inverse integrating factor} of system (1) if it satisfies \(\mathcal{X} u = u (\text{div} \mathcal{X})\). In the present paper the authors give some sufficient conditions on the existence of the inverse integrating factors for certain classes of \(n\)-th order autonomous differential systems giving the explicit forms of the inverse integrating factors. The results are applied to several examples.
\(n\)-order autonomous differential systems
scientific article; zbMATH DE number 6370926
scientific article