Modelling a spring system with damping force and external forces

[sss1]

Homework Statement

I know for a system with no external forces there are conditions for being underdamping, overdamping and critically damped. Is there also such conditions for systems having external forces acting on them also? Specifically, for the example 10y''+9y"+2y'=-2e^(-t/2) with conditions y(0)=0 and y'(0)=0, is the system critically damped?

Relevant Equations

10y''+9y"+2y'=-2e^(-t/2)

I think its critically damped by looking at the graph of the solution.

 

[kuruman]

Can you give an example of a system with no external forces that exhibits damping and a system with external forces that also exhibits damping? I do not understand your use of "external force" at least not in the Newtonian sense.

Also, if 10y''+9y"+2y'=-2e^(-t/2), why not 19y''+ 2y'=-2e^(-t/2)? Is there a real difference between y'' and y"?