In a physical problem I need to investigate the following nonlinear differential equation

$$\ddot x+\omega^2\left (1+\frac{m^2\dot x^2}{p^2}\right)x=0,$$

where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:


This type of ODE, $$\ddot{x}+f(x)\dot{x}^2+g(x)=0$$ is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).

A particular case with nice properties is $$\ddot{x}-\frac{f'(x)}{f(x)}\dot{x}^2+f(x)\int_0^x\frac{1}{f(u)}du=0,$$ see Design of nonlinear isochronous oscillators.


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