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Consider the equation of motion #+ 7 =0 (1) (y is a positive constant) with initial conditions *=T, and =0 at t = 0. (2) (a) Show that the period of the oscillations is T = \\ 2 B ( +. 1) - 7.4163 1 (3) Here, B denotes the beta function B(+ +) = 1-3/4(1 -u)-1/2du * 5.2441. (4) (b) Show that the Fourier expansion for a(t) can be expressed as x(t) = > dan+1 cos(2n + 1 )wt (w = 27 /T) . (5) n=0 (c) Use an integration by parts (twice) to express the Fourier coefficients 2w / W r(t) cos nut dt (6) as 2w/ ON = ' (t) cos nut dt . (7) (d) Use the ansatz r(t) = a, cos wt (8) in (7) to obtain the estimates a, = 2w/ V37, da = 0, /27 ~ 0.03704a, . (9) Impose the initial condition (2), on the corresponding approximate solution r = a, cos wt + 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 = * cos wt + + cos 3wt.) (11)