The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

中文翻译：

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二次阻尼强非线性达芬振荡器的精确解

在经典动力学的中间教学中很少涉及非线性运动方程，例如Duffing振荡器方程及其族。这是有问题的，因为以前无法​​通过基本功能解决它。因此，在这项工作中，研究了二次阻尼高阶非线性Duffing振荡器的稳定性。在下文中，获得了Weierstrass椭圆函数形式的无阻尼高阶非线性Duffing振荡器的一些新的解析解。随后，首次详细推导了在特定条件下（不是任意初始条件）并且呈Weierstrass椭圆函数形式的二次阻尼高阶非线性Duffing方程的精确解析解。此外，