# Overdamping condition

In case of equation
$$\alpha y''(x)+\beta y'(x)+\gamma y(x)=0$$
where ##\alpha>0##, ##\beta>0##, ##\gamma>0##, characteristic equation is
$$\alpha r^2+\beta r+\gamma=0$$
and characteristic roots are
$$r_{1,2}=\frac{-\beta \pm \sqrt{\beta^2-4\alpha \gamma}}{2 \alpha}$$
If ## \beta^2<4\alpha \gamma## system is underdamped, and
if ## \beta^2>4\alpha \gamma## system is overdamped.
What in the case of equation
$$\alpha y''(x)+\beta y'(x)+\gamma \sin[y(x)]=0$$
when equation is nonlinear? How to find when system is overdamped? Thanks a lot for your help in advance.

## Answers and Replies

haushofer
How do you define underdamping in such a situation? Is such an equation even solvable? And does it have periodic solutions (i.e. imaginary exponential parts in its solution)?

Overdamping is when characteristic roots are real and negative, I suppose. But I am not sure how to see that in case of nonlinear equations. I saw in literature that people discuss overdamped limit in case of nonlinear equations, but I am not sure how to do that. Because of that I asked here in the forum.

This limit is to my mind important to see when term ##\beta y'(x)## dominates over ##\alpha y''(x)##.

And just one more question, but very similar that the mentioned. What if we have equation
$$\alpha y''(x)+\beta y'(x) +\gamma y(x)=cz+d$$
If ##c## and ##d## are not functions of ##x##? What's happening with condition in that case?