ps6sol - 18.03 Problem Set 6 Solutions 22[Part I problems 4...

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18.03 Problem Set 6: Solutions 22. [Part I problems 4] (a) [8] g ( t ) = π 2 - 4 π (cos( t )+ cos(3 t ) 3 2 + · · · ) according to L21 or II.21(f). Since x p = A cos( ωt ) ω 2 n - ω 2 is a solution to ¨ x + ω 2 n x = A cos( ωt ), by Superposition III x p = π 2 ω 2 n - 4 π cos( t ) ω 2 n - 1 + cos(3 t ) 3 2 ( ω 3 n - 3 2 ) + · · · is a solution to ¨ x + ω 2 n x = g ( t ) (provided none of the denominators vanishes). (b) [4] The system is in resonance for ω n = 1 , 3 , 5 , . . . . (c) [4] For ω n near to 1, the dominant term in x p is - 4 π cos( t ) ω 2 n - 1 . This is a large positive multiple of cos( t ) for ω n < 1 and a large negative multiple of cos( t ) for ω n > 1. (d) , (e) [4] When ω n is an integer, the homogeneous solutions A cos( ω n t - φ ) have period 2 π , so if some particular solution is periodic of period 2 π then all solutions are. This occurs when ω n is an even integer. More generally, if the particular solution is periodic (necessarily of period 2 π ) and some integer multiple of its period coincides with an integer multiple of the period of the nonzero homogeneous solutions ( 2 π ω n ), then all solutions are periodic of that common period. This happens when ω n
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