On a generalization of the Chazy equation with a movable singular line
The differential equation \[ y^{(2n + 1)} = 2y y^{(2n)} + \sum_{m = 1}^n a_m y^{(m)} y^{(2n - m)} \tag{1} \] is considered, where \[ {a_m} = \frac{( - 1)^m}{n + 1}\left( \begin{matrix} 2n \\ m \end{matrix} \right)\left( \begin{matrix} 2n + 2 \\ m + 1 \end{matrix} \right),\qquad m = 1,\dots,n - 1;\;a_n = \frac{( - 1)^n(2n + 1)}{(n + 1)^2}\left( \begin{matrix} 2n \\ n \end{matrix} \right). \] It is possible to consider equation (1) as a generalisation of the Chazy equation as it keeps its specific features. For example, the equation (1) has only negative resonances and it has a movable singular line. The authors derive some general results for the equation (1). For example: {\parindent7mm \begin{itemize}\item[(i)] Let \(c_1, \dots, c_{2n + 1}\) be arbitrary constants. Then here is a solution of the equation such that \[ y = - (n + 1)(2n + 1)\left((z - z_0)^{ - 1} + \sum_{k=1}^\infty {{\alpha _k} \over {(z - z_0)^{k + 1}}} \right) \] with \(\alpha _k=c_k\), \(k=1,\dots , 2n+1\); \item[(ii)] Let \(c_1, \dots, c_n\) be arbitrary constants. Then \[ y = - (n + 1)(2n + 1)\left( (z - z_0)^{ - 1} + \sum_{k=1}^n {{c_k} \over {(z - z_0)^{k + 1}}} \right) \] is a rational solution of (1). \end{itemize}}
scientific article; zbMATH DE number 5659355
scientific article