I'm considering a damped harmonic oscillator $$\ddot{x}+\alpha\dot{x}+\omega^2x=0 \,\,\,\,\,\,\,\,\,\,\, \alpha \neq 0$$ I know that this equation could not be a lagrangian equation originated from a lagrangian L$(x,\dot{x},t)$.
Reading about this problem I find that I can determinate a function $f(t)>0$ such as $$f(t)(\ddot{x}+\alpha\dot{x}+\omega^2x)=0$$ is a lagrangian equation and for this reason I can determinate a lagrangian.
However I don't kwon how to do it. I need an analitical expression for the function $f(t)$ ed the lagrangian.
