# Nonlinear oscillator with velocity-dependent frequency

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.