Explanation Given: The boundary conditions are as follows: u ( 0 , t ) = 0 E ∂ u ∂ x | x = L = F 0 or E u x ( L , t ) = F 0 u ( x , 0 ) = 0 ∂ u ∂ t | t = 0 = 0 or u t ( x , 0 ) = 0 Calculation: Take Laplace transform on both sides of equation a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 as follows: a 2 L { ∂ 2 u ∂ x 2 } = L { ∂ 2 u ∂ t 2 } a 2 d 2 U d x 2 = s 2 U ( x , s ) − s u ( x , 0 ) − u t ( x , 0 ) ...... ( 1 ) Take Laplace transform of boundary condition u ( 0 , t ) = 0 as follows: L { u ( 0 , t ) } = L { 0 } U ( 0 , s ) = 0 ...... ( 2 ) Take Laplace transform of boundary condition E u x ( L , t ) = F 0 as follows: L { E u x ( L , t ) } = L { F } E U x ( L , s ) = F U x ( L , s ) = F E ...... ( 3 ) Substitute u ( x , 0 ) = 0 and u t ( x , 0 ) = 0 in equation (1). a 2 d 2 U d x 2 = s 2 U ( x , s ) − s u ( x , 0 ) − u t ( x , 0 ) a 2 d 2 U d x 2 = s 2 U ( x , s ) d 2 U d x 2 − s 2 a 2 U ( x , s ) = 0 Solve the differential equation by the method of undetermined coefficient. The auxiliary equation for the above differential equation as follows: r 2 − s 2 a 2 = 0 Solve the above auxiliary equation as follows: r 2 − s 2 a 2 = 0 r 2 = s 2 a 2 r = ± s a ( Square root ) Therefore, the complementary function of the differential equation d 2 U d x 2 − s 2 a 2 U ( x , s ) = 0 is shown below. C .F = A cosh s a x + B sinh s a x The particular integral of the differential equation d 2 U d x 2 − s 2 a 2 U ( x , s ) = 0 is obtained as follows: ( D 2 − s 2 a 2 ) U ( x , s ) = 0 P .I = 1 D 2 − ( s / a ) 2 ( 0 ) ( P .I = U ( x , s ) ) P .I = 0 Therefore, the solution of the differential equation d 2 U d x 2 − s 2 a 2 U ( x , s ) = 0 is as follows: U ( x , s ) = C .F + P .I U ( x , s ) = A cosh s a x + B sinh s a x

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 14.2, Problem 7E

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The longitudinal displacement u(x,t) for a cross section of the bar subjected to wave equation a2∂2u∂x2=∂2u∂t2.

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Chapter 14.2, Problem 6E

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Chapter 14 Solutions

Differential Equations with Boundary-Value Problems

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

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The longitudinal displacement u ( x , t ) for a cross section of the bar subjected to wave equation a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 .

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The longitudinal displacement u(x,t) for a cross section of the bar subjected to wave equation a2∂2u∂x2=∂2u∂t2.

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Chapter 14 Solutions