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s11_mthsc208_hw17 - MthSc 208 Fall 2011(Differential...

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MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 17 Due Friday April 14th, 2011 (1) Solve the following differential equations: (a) y 0 = ky (b) y 0 = - ky (c) y 00 = k 2 y (d) y 00 = - k 2 y (e) y 00 + 3 y 0 + 2 y = 0 (f) y 00 + 2 y 0 + 2 y = 0 (g) y 00 + 2 y 0 + y = 0 (2) Suppose that f is a function defined on R (not necessarily periodic). Show that there is an odd function f odd and an even function f even such that f ( x ) = f odd + f even . Hint : As a guiding example, suppose f ( x ) = e ix , and consider cos x = 1 2 ( e ix + e - ix ) and i sin x = 1 2 ( e ix - e - ix ). (3) Express the y -intercept of f ( x ) = a 0 2 + X n =1 a n cos nx + b n sin nx in terms of the a n ’s and b n ’s. ( Hint : It’s not a 0 or a 0 / 2!) (4) Consider the 2 π -periodic function f ( x ) = a 0 2 + X n =1 a n cos nx + b n sin nx . Write the Fourier series for the following functions: (a) The reflection of f ( x ) across the y -axis; (b) The reflection of f ( x ) across the x -axis; (c) The reflection of f ( x ) across the origin. (5) (a) The Fourier series of an odd function consists only of sine-terms. What additional symmetry conditions on f will imply that the sine coefficients with even indices will be zero (i.e., each b 2 n = 0)? Give an example of a non-zero function satisfying this additional condition.
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