Answer Happy

Consider a driven, damped harmonic oscillator modeled by the ODE dax dc + 5 + 4x = cos 3t dt dt2 (a) Write an expression for the general solution to the complementary homogeneous equation, xe(t), in terms of two arbitrary constants A and B. Explain whether this is an under- damped, over-damped, or critically-damped oscillator. (b) Explain why an ansatz (guess) of the form xp(t) = C sint + D cost will not solve the inhomogeneous equation for any possible combinations of C and D. (c) Using the ansatz wp(t) = C sin 3t + D cos 3t, find a particular solution to the inhomogeneous equation. (d) Write an expression for the general solution to the inhomogeneous equation. (Recall how many arbitrary constants a general solution is allowed to have. Your solution should not have more arbitrary constants than is allowed.) (e) Find the unique solution to the inhomogeneous equation with initial conditions x(0) = 1 and :(0) = 0.