� � � � � � � � � � 7. Fourier Series Based on exercises in Chap. 8, Edwards and Penney, Elementary Differential Equations 7A. Fourier Series 7A-1. Find the smallest period for each of the following periodic functions: a) sin �t/3 b) | sin t| c) cos2 3t 7A-2. Find the Fourier series of the function f(t) of period 2� which is given over the interval −� < t � � by 0, −� < t � 0; −t, −� < t < 0; a) f(t) = b) f(t) = 1, 0 < t � � t, 0 � t � � 0, m �= n; 7A-3. Give another proof of the orthogonality relations cos mt cos nt dt = −� �, m = n . 1by using the trigonometric identity: cos A cos B = 2 cos(A + B) + cos(A − B) . 7A-4. Suppose that f (t) has period P . Show that I f(t) dt has the same value over any interval I of length P , as follows: � a+P a a) Show that for any a, we have f(t) dt = f(t) dt. (Make a change of variable.) P 0 � P� a+P b) From part (a), deduce that f(t) dt = f(t) dt. a 0 7B. Even and Odd Series; Boundary-value Problems 7B-1. a) Find the Fourier cosine series of the function 1 − t over the interval 0 < t < 1, and then draw over the interval [−2, 2] the graph of the function f(t) which is the sum of this Fourier cosine series. b) Answer the same question for the Fourier sine series of 1 − t over the interval (0, 1). 7B-2. Find a formal solution as a Fourier series, for these boundary-value problems (you can use any Fourier series derived in the book’s Examples): a) x�� + 2x = 1, x(0) = x(�) = 0; b) x�� + 2x = t, x�(0) = x�(�) = 0 (use a cosine series) � 0 � a 7B-3. Assume a > 0; show that f(t) dt = ± f(t) dt, according to whether f(t) −a 0 is respectively an even function or an odd function. 7B-4. The Fourier series of the function f(t) having period 2, and for which f (t) = t2 for 0 < t < 2, is 4 4 � cos n�t 4 � sin n�t f(t) = + − . 3 �2 n2 � n 1 1 Differentiate this series term-by-term, and show that the resulting series does not converge to f�(t). 12 18.03 EXERCISES 7C. Applications to resonant frequencies 7C-1. For each spring-mass system, find whether pure resonance occurs, without actually calculating the solution. a) 2x�� + 10x = F (t); F (t) = 1 on (0, 1), F (t) is odd, and of period 2; b) x�� + 4�2x = F (t); F (t) = 2t on (0, 1), F (t) is odd, and of period 2; c) x�� + 9x = F (t); F (t) = 1 on (0, �), F (t) is odd, and of period 2�. 7C-2. Find a periodic solution as a Fourier series to x�� + 3x = F (t), where F (t) = 2t on (0, �), F (t) is odd, and has period 2�. 7C-3. For the following two lightly damped spring-mass systems, by considering the form of the Fourier series solution and the frequency of the corresponding undamped system, determine what term of the Fourier series solution should dominate — i.e., have the biggest amplitude. a) 2x�� + .1x� + 18x = F (t); F (t) is as in 7C-2. b) 3x�� + x� + 30x = F (t); F (t) = t − t2 on (0, 1), F (t) is odd, with period 2.MIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations�� Spring 2010 For information about citing these materials or our Terms of Use, visit:
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