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Consider the equation of motion + + 7 =0 (1) (y is a positive constant) with initial conditions dan+: cos(2n + 1)wt (w = 2#/ T) . (5) 1=0 (c) Use an integration by parts (twice) to express the Fourier coefficients a. = a(t) cos nut dt (6) as ON = - a' (t) cos nut dt . (7) (d) Use the ansatz r(t) = a, coswt (8) in (7) to obtain the estimates a, = 2w/ V37, da = a, /27 0.03704a, . (9) Impose the initial condition (2), on the corresponding approximate solution x = a, coswt + a, cos 3wt and deduce that this yields T, 1.01AT (10) for the period, where T is the actual value (3). (Hint: Use the identity cos' wt = * coswt + + cos 3wt.) (11) (e) Repeat (d), starting with the more accurate approximation x(t) = a, coswt + a, cos 3wt, (12) instead of (8). Show that this yields a, ~ 0.9782(2w/V37), as ~0.0450la, , as ~ 0.001723a, , (13) and an improved estimate for the period T, & 1.002 T. (14) (Hint: Use also the identity cos' wt cos 3wt = + coswt + } cos 3wt + + cos 5wt.) (15)