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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 20 Due Monday April 6th, 2009 (1) Determine which of the following functions are even, which are odd, and which are neither even nor odd: (a) f ( t ) = t 3 + 3 t . (b) f ( t ) = t 2 + | t | . (c) f ( t ) = e t . (d) f ( t ) = 1 2 ( e t + e- t ). (e) f ( t ) = 1 2 ( e t- e- t ). (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 : Look at Problem (1c), (d), and (e) together. (3) (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. (b) What symmetry conditions on f will imply that the sine coefficients with odd indices will be zero (i.e., each b 2 n +1 = 0)? Give an example of a non-zero function satisfying= 0)?...
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