Unformatted text preview: MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 16 Due Friday April 8th, 2011 (1) Solve the following differential equations: (a) y  2y = 0 (a) y  2y = t  3 (b) y  2y = e3t (c) y  4y = 0 (d) y + 4y = 0 (e) y + 4y + 3y = 10. (2) The function  x < /2, 0 1 /2 x < /2, f (x) = 0 /2 x , can be extended to be periodic of period 2. Sketch the graph of the resulting function, and compute its Fourier series. (3) The function f (t) = x, for x [, ] can be extended to be periodic of period 2. Sketch the graph of the resulting function, and compute its Fourier series. (4) The function 0  x < 0, f (x) = x 0 x , can be extended to be periodic of period 2. Sketch the graph of the resulting function, and compute its Fourier series. (5) Consider the 2periodic function defined by f (x) = x2 f (x  2k),  x < ,  + 2k x < + 2k. Sketch this function (at least for k = 2, 1, 0, 1, 2) and compute its Fourier series. (6) Find the Fourier series of the following functions without computing any integrals. (a) f (x) = 2  3 sin 4x + 5 cos 6x, (b) f (x) = sin2 x [Hint: Use a standard trig identity.] (7) Determine which of the following functions are even, which are odd, and which are neither even nor odd: (a) f (t) = x3 + 3x. (b) f (t) = x2 + x. (c) f (t) = ex . (d) f (t) = 1 (ex + ex ). 2 (e) f (t) = 1 (ex  ex ). 2 ...
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 Spring '09
 Staufeneger
 Differential Equations, Equations, Derivative, Fourier Series, 2k, Dr. Macauley HW

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