hmk3_sol - Math 267 – Assignment 3 1. (a) Given an...

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Unformatted text preview: Math 267 – Assignment 3 1. (a) Given an integer k , find the constant d k for which y k ( t ) = d k e ikt is a particular solution of the DE y ( t ) + 2 y ( t ) + 3 y ( t ) = e ikt . (b) Suppose that f ( t ) has a Fourier series expansion f ( t ) = ∞ k =-∞ c k e itk . Using the result from (a), find a particular solution to y ( t ) + 2 y ( t ) + 3 y ( t ) = f ( t ) . Solution. (a) We have y k ( t ) = d k e ikt , y k ( t ) = ( ik ) d k e ikt and y k ( t ) =- k 2 d k e ikt . We substitute into the DE and get:- k 2 d k e ikt + 2( ik ) d k e ikt + 3 d k e ikt = e ikt . This implies (- k 2 + 2( ik ) + 3) d k = 1 , and therefore d k = 1- k 2 + 2( ik ) + 3 . (Note that for all k the denominator is never zero) (b) Using the result of ( a ) and superposition, we see that y ( t ) = ∞ k =-∞ c k d k e itk = ∞ k =-∞ c k e itk- k 2 + 2( ik ) + 3 . is a particular solution. 2. Suppose that f ( t ) is a 2 π-periodic function whose nonzero Fourier coeffi- cients (in complex form) are given by...
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This note was uploaded on 02/02/2010 for the course MATH MATH 267 taught by Professor Phan during the Spring '10 term at The University of British Columbia.

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hmk3_sol - Math 267 – Assignment 3 1. (a) Given an...

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