I Overdamping condition

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

haushofer

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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
[tex]\alpha y''(x)+\beta y'(x) +\gamma y(x)=cz+d[/tex]
If ##c## and ##d## are not functions of ##x##? What's happening with condition in that case?
 

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