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Name________________________ Student I.D.___________________ Math 2280-001 Quiz 12 April 24, 2015 You may choose to do #1 (Laplace transform), or #2 (Fourier series). Indicate clearly which problem you'd like graded. 1) Find a formula for solutions to the forced oscillation problem below, as convolution integrals involving the forcing function f and the "weight function". Hint: when you take the Laplace transform of the differential equation and solve for X s if will be of the form X s = F s G s . x## t C 2 x# t C 10 x = f t x 0 =0 x# 0 = 0 (10 points) 2) Find the Fourier series for the square wave of period P = 2 L = 4 and amplitude 3. In other words, for the period 4 extension of K3 K2 ! t ! 0 f t = . 3 0!t!2 You may do this directly from the definition N N a0 p p fw C an cos n t C bn sin n t 2 L L n=1 n=1 with > > L 1 a0 = L an d bn d f, cos n f, sin n p t L = p t L = g t dt KL L 1 L f t cos n p t dt, n 2 ; L f t sin n p t dt, n 2 ; L KL L 1 L KL or by rescaling the 2 pKperiodic unit square wave function satisfying square t = K1 Kp ! t ! 0 0!t!p 1 which has Fourier series square t = 4 p > n odd 1 sin n t n (10 points)