Homework 4 - Homework Set 4 Due Monday October 8 2012 1...

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Homework Set 4 Due: Monday, October 8, 2012 1. Find the Fourier coefficients a n and b n for the function shown below. 0 1 2 3 4 f max Using Mathematica , make a plot comparing the function itself with the first couple of terms in the Fourier series, and another for the first six or so terms. Take f max = 1. 2. Consider the nonlinear first-order equation ˙ x = 2 x - 1. ( a ) Separating variables, find a solution x 1 ( t ). ( b ) Your solution should contain one constant of integration k , so you might reasonably expect it to be the general solution. Show, however, that there is another solution x 2 ( t ), that is not of the form of x 1 ( t ) whatever the value of k . ( c ) Show that although x 1 ( t ) and x 2 ( t ) are solutions, neither Ax 1 ( t ), nor Bx 2 ( t ), nor x 1 ( t ) + x 2 ( t ) are solutions. (That is, the superposition principle does not apply to this equation.) 3. Lord Rayleigh used the equation ¨ x - ( a - b ˙ x 2 ) ˙ x + ω 2 0 x = 0 in his discussion of non-linear effects in acoustics.
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