Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on such transformations is the class of linearisable second-order ordinary differential equations (ODEs). There are various characterisations of such ODEs. We exploit a particular characterisation and the expanded Lie group method to construct a generic solution for all linearisable second-order ODEs. The general solution of any given equation from this class is then easily obtainable from the generic solution through a point transformation constructed using only two suitably chosen symmetries of the equation. We illustrate the approach with three examples.

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关于通过点变换解线性化二阶常微分方程的一些说明

在求解复杂方程的许多例程中，微分方程到其他等效方程的转换都扮演着核心角色。一类特别适用于基于此类变换的求解技术的微分方程是一类可线性化的二阶常微分方程（ODE）。这种ODE具有多种特征。我们利用特定的特征和扩展的李群方法来构造所有线性化二阶ODE的通用解决方案。从这个类任何给定的方程的一般解决方案是然后从通过仅使用两个等式的适当选择的对称性构造的点变换的一般的解决方案可以容易地获得。我们通过三个示例说明该方法。