3. Suppose we have a driven damped oscillator modeled by y + 2y + 4y = 3 cos(t).

(a) Find the steady-state solution for the oscillator.

(b) Rewrite the answer to part (a) in phase-amplitude form. How does your amplitude relate to the gain function for the system? (c) Plot the amplitude for your steady-state solution as a function of for = 0.25, 0.1, and = 0. What do these plots tell you about how the oscillator responds to the driving function 3 cos(t)

(d) Suppose = 0 and = 2. Can we determine the long-term behavior using the plots from part (c)? Find the particular solution in this case and compare it to what you predicted.