In this paper, we study a nonlinear second-order ordinary differential equation which we call the Ermakov–Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced either to the Ermakov or the Painlevé XXV equation. The Ermakov–Painlevé XXV equation arises from a generalized Riccati equation and the related third-order linear differential equation via the Schwarzian derivative. The generalized Riccati equation has two families of Riccati solutions and we study the corresponding solutions to the Ermakov–Painlevé XXV equation. We show that one of these families appears only in the Ermakov case. We give examples of the Ermakov–Painlevé XXV equations and show how to construct their solutions expressed in terms of elementary or in terms of the classical special functions.

在本文中，我们研究了一个非线性二阶常微分方程，我们将其称为 Ermakov-Painlevé XXV 方程，因为在其系数的某些限制下，它可以简化为 Ermakov 或 Painlevé XXV 方程。Ermakov-Painlevé XXV 方程源自广义 Riccati 方程和通过 Schwarzian 导数相关的三阶线性微分方程。广义 Riccati 方程有两个 Riccati 解族，我们研究了 Ermakov-Painlevé XXV 方程的相应解。我们表明这些家庭之一仅出现在叶尔马科夫案例中。我们给出了 Ermakov-Painlevé XXV 方程的示例，并展示了如何构造用初等函数或经典特殊函数表示的解。