For ODE y\"+2dy\'+y=cost, let d be a positive, but adjustable, parameter. Descir

Question

For ODE y"+2dy'+y=cost, let d be a positive, but adjustable, parameter. Descirbe the behavior of the steady state solution as d-->0+. Interpret this in regards to the concepts of pure versus practical resonance.

Explanation / Answer

When d>0 the system is a forced, damped oscillator. The forcing frequency is 1, which is the same as the natural frequency of the corresponding undamped system. When d is small (but nonzero) there is a little bit of damping that will result is a large, but not infinite, response. This is practical resonance. When d = 0 the system is an undamped, forced oscillator. Since the forcing frequency is the same as the natural frequency of the corresponding undamped system, the system will exhibit pure resonance; that is the response y(t) will approach infinity as gets large.

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