论文标题
关于通过非局部变换的线性化性和二阶普通微分方程的第一积分
On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations
论文作者
论文摘要
非线性二阶普通微分方程在各个科学领域（例如物理学，力学和生物学）很常见。在这里，我们通过通过某些非局部转换考虑线性问题的一般情况来提供一个新的可集成二阶普通微分方程的家族。此外，我们表明，当第一个积分是自主或理性的时，来自可线化家族的每个方程都承认了先验的第一积分和研究特殊情况。因此，作为解决这个线性化问题的副产品，我们获得了承认某个先验第一积分的二阶微分方程的分类。为了证明我们的方法的有效性，我们考虑了几个自主和非自主二阶微分方程的例子，包括对行驶和范德波尔振荡器的概括，以及构建其第一个积分和一般解决方案。我们还表明，相应的第一积分可用于查找所考虑方程的周期性解决方案，包括极限周期。
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.
