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Math · Advanced Math
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b) [30%] Consider the non-smooth differential equation: dt2 where f(y) -1 for у > 0, = and f(y) + 1 for y < 0. Determine the solution y(t) to this differential equation given initial condition y y = 1 at t = O. Sketch the solution y(t) from time t = 0 to t = 2. 1 andd) (10 %) Say you are given a well-behaved function p(1) in the interval 0 < t < 2π. It is not known what the value of the function is outside of this interal. In particular, you do not know if it is periodic. In a couple of sentences, outline how you would find a Fourier sine series2 that converges to p(t) in the interval (0, 2T) A Fourier cosine series has only cosine terms, no sine terms A Fourier sine series has only the sine terms, no cosine terms. System Dynamics & Vibration Ph.D. Qualifying Examination Student ID # roblem # M11 (contd) Is this Fourier sine series unique? (That is, is there only one way to get a Fourier sine series and only one Fourier sine series that converges to the function in the given interval?) e) (5 %) For the function p(t) from part-d, is it also possible to find a Fourier cosine series that converges to p(t) in the interval (0, 2)? If no, why? If yes, how? f) (5 %) What is the period of the periodic function/2(t) given by: 2(t)-4cos 35mt +3sin 2

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