hw_4_2_3_09

hw_4_2_3_09 - MA 221 Homework Solutions Due date: April...

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MA 221 Homework Solutions Due date : April 2 / 3 , 2009 Sec 10.2 Problems #1 ̄ 9 ̄ ,2 ̄ 0 ̄ ̄ 2 ̄ Sec 10.6 Problem #3 ̄ Section 10 . 2 Solve the vibrating string problem with 3, L , and the given initial functions f x and g x . 1 ̄ 9 ̄ .) f x 3sin2 x 12sin13 x , g x 0 By letting 3and L in formula (24) on page 585 of the text, we see that the solution we want will have the form u x , t n 1 a n cos3 nt b n sin3 nt sin nx Therefore, we see that u t n 1 3 na n nt 3 nb n nt sin nx In order for the solution to satisfy the initial conditions, we must find a n and b n such that u x ,0 n 1 a n sin nx x x ,( 1 ) and u x t n 1 3 nb n sin nx 0. From the first condition, we observe that we must have a term for n 2,13 and for these terms we want a 2 a 13 12. All of the other a n ’s must be zero. By comparing coefficients in the second condition, we see that all b n ’s must also be zero. Therefore, by substituting these values into equation (1) above, we obtain the solution of the vibrating string problem with 3, L and f x and g x as given. This solution is given by u x , t 3 cos 3  2 t sin2 x 12 cos 3  13 t sin13 x 0 Or by simplifying, we obtain u x , t 3cos6 t x 12cos39 t x 2 ̄ 0 ̄ .) f x 0, g x 2sin3 x 9sin7 x sin10 x By letting L in formula (24) on page 585 of the text, we see that the solution we want will have the form u x , t n 1 a n nt b n nt sin
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hw_4_2_3_09 - MA 221 Homework Solutions Due date: April...

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