MATH 285 E1 F1 GRADED HOMEWORK SET 4 DUE FRIDAY OCTOBER 24 IN LECTURE This time the homework has just one part Please staple your homework together and put your name and section on it Thank you 1 10 points Find the general solution of the differential equation D 3 y xe2x e3x Hint Use the annihilator method 2 5 points Consider the forced mass spring oscillator with mass m 4 no damping c 0 and a spring constant k that is adjustable Consider a driving forces that is the sum of two cosine functions F t cos 2t cos 5t The differential equation for the displacement x t is then 4x00 kx cos 2t cos 5t For what values of k is it impossible to find a particular solution of the form A cos 2t B cos 5t These are the values of k for which resonance occurs 3 5 points Recall from the lecture the damped c 0 forced oscillator obeying the differential equation mx00 cx0 kx F0 cos t In the lecture we derived a particular solution of this equation xp t C cos t where F0 C p k m 2 2 c 2 tan c k m 2 Here is something we did not talk about in the lecture the solution xp t is called the steady state solution because any solution will asymptotically converge to it as t More precisely if x1 t is any solution of the same nonhomogeneous equation then lim x1 t xp t 0 t Explain why this is true You should use the description of the solutions of the homogeneous equation mx00 cx0 kx 0 on page 191 of the textbook Hint It is very important that c 0 1 2 GRADED HOMEWORK 4 4 5 points Use the orthogonality properties of sine and cosine to compute the following integral Z 5 sin 2t cos 3t cos 2t 5 sin 6t sin 2t dt
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