Explanation Given: The given initial value problem is, ∂ 2 u ∂ t 2 = 16 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 u ( 0 , t ) = u ( π , t ) = 0 , t > 0 u ( x , 0 ) = sin 2 x , 0 < x < π ∂ u ∂ t ( x , 0 ) = 1 − cos x , 0 < x < π Approach: The motion of the string initial boundary value problem is given by, ∂ 2 u ∂ t 2 = α 2 ∂ 2 u ∂ x 2 , 0 < x < L , t > 0 u ( 0 , t ) = u ( L , t ) = 0 , t > 0 u ( x , 0 ) = f ( x ) , 0 < x < L ∂ u ∂ t ( x , 0 ) = g ( x ) , 0 < x < L The formal solution is determined by method of separation of variables. u ( x , t ) = ∑ n = 1 ∞ [ a n cos n π α L t + b n sin n π α L t ] sin n π x L … ( 1 ) Here, a n = 2 L ∫ 0 L f ( x ) sin n π L x d x ( 2 ) b n = ( L n π ) 2 L ∫ 0 L g ( x ) sin n π L x d x ( 3 ) f ( x ) = ∑ n = 1 ∞ a n sin n π x L g ( x ) = ∑ n = 1 ∞ b n ( n π α L ) sin n π x L Calculation: Consider the given initial value problem. ∂ 2 u ∂ t 2 = 16 ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 u ( 0 , t ) = u ( π , t ) = 0 , t > 0 u ( x , 0 ) = sin 2 x , 0 < x < π ∂ u ∂ t ( x , 0 ) = 1 − cos x , 0 < x < π Compare the above initial value problem with motion of the string initial boundary value problem. α = 4 L = π f ( x ) = sin 2 x g ( x ) = 1 − cos x Substitute 4 for α and π for L in equation ( 1 ) . u ( x , t ) = ∑ n = 1 ∞ [ a n cos 4 n t + b n sin 4 n t ] sin n x Substitute π for L and sin 2 x for f ( x ) in equation ( 2 ) . a n = 2 π ∫ 0 π sin 2 x sin n π x d x = 1 π ∫ 0 π ( 1 − cos 2 x ) sin n x = 1 π ∫ 0 π ( sin n x − cos 2 x sin n x ) = 1 π [ [ − cos n x n ] 0 π + 1 2 ∫ 0 π sin ( n + 2 ) x + sin ( n − 2 ) x d x ] Simplify the above equation. a n = 1 π [ [ − cos n x n ] 0 π + 1 2 [ cos ( n + 2 ) x ( n + 2 ) + cos ( n − 2 ) x ( n − 2 ) ] 0 π ] = 1 π [ − 1 n ( cos n π − 1 ) + 1 2 [ ( − 1 ) n + 2 ( n + 2 ) − 1 ( n + 2 ) + ( − 1 ) n − 2 ( n − 2 ) − 1 ( n − 2 ) ] ] = 1 π [ − 1 n ( ( − 1 ) n − 1 ) + 1 2 [ 1 ( n + 2 ) ( ( − 1 ) n − 1 ) + 1 ( n − 2 ) ( ( − 1 ) n − 1 ) ] ] = 1 π [ − 1 n + n n 2 − 4 ( ( − 1 ) n − 1 ) ] a n = { 2 π ( n n 2 − 4 − 1 n ) , when n is odd 0 , when n is odd , Substitute π for L and ( 1 − cos x ) for g ( x ) in equation ( 3 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Fundamentals of Differential Equations and Boundary Value Problems

Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - Prob. 6E Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 1-8, determine all the solutions, if...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...Ch. 10.2 - In Problems 9-14, find the values of eigenvalues...

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The formal solution of initial boundary problem.

Solution Summary: The author explains that the formal solution of initial boundary problem is u(x,t)=displaystyle undersetn=1

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The formal solution of initial boundary problem.