s11_mthsc208_hw17

s11_mthsc208_hw17 - MthSc 208, Fall 2011 (Differential...

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Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 17 Due Friday April 14th, 2011 (1) Solve the following differential equations: (a) y = ky (b) y =- ky (c) y 00 = k 2 y (d) y 00 =- k 2 y (e) y 00 + 3 y + 2 y = 0 (f) y 00 + 2 y + 2 y = 0 (g) y 00 + 2 y + 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 2 + X n =1 a n cos nx + b n sin nx in terms of the a n s and b n s. ( Hint : Its not a or a / 2!) (4) Consider the 2 -periodic function f ( x ) = a 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...
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