3. Review: Continue with HW7, Q5(b), where we considered the differential equation with 0 < b < 2w0,

y"+by+wo^2y = cos(wt),

which represents the movement of a harmonic oscillator with under-damping. We know

Questions:

(a) Discuss the behavior of yp(t) when w=0. In particular, what's A, a, and yp(t)?

(After this special case, we only look at w€ (0,00) below.)

(b) Discuss the behavior of yp(t) when w=wo.

(*) (you don't need to submit this part)

Plot the A(w) graph on Desmos. Observe where the maximum occurs over w€ (0,00). Start with

1, and try more sets of parameter values. Keep in mind that this is the case of under-damping:

0

(c) Find the precise w value where the maximum value of A(w) occurs over w€ (0,00). Hint: this is a

calculus 1 problem on the function g(w) = (w² - w²)² + b²w². Second derivative test is the right tool to

= ليا = b

=

use.

Is your finding compatible with your observations in part (*) above?