Explanation Given: The boundary conditions are as follows: u ( 0 , t ) = 60 + 40 U ( t − 2 ) lim x → ∞ u ( x , t ) = 60 u ( x , 0 ) = 60 Solution: Take Laplace transform on both sides of equation u x x = u t . L { u x x } = L { u t } d 2 U d x 2 = s U ( x , s ) − u ( x , 0 ) ...... ( 1 ) Take Laplace transform of boundary condition u ( 0 , t ) = 60 + 40 U ( t − 2 ) as follows: L { u ( 0 , t ) } = L { 60 + 40 U ( t − 2 ) } U ( 0 , s ) = 60 L { 1 } + 40 L { U ( t − 2 ) } U ( 0 , s ) = 60 s + 40 e − 2 s s ...... ( 2 ) Take Laplace transform of boundary condition lim x → ∞ u ( x , t ) = 60 as follows: L { lim x → ∞ u ( x , t ) } = L { 60 } lim x → ∞ U ( x , s ) = 60 s ...... ( 3 ) Substitute u ( x , 0 ) = 60 in equation (1). d 2 U d x 2 = s U ( x , s ) − 60 d 2 U d x 2 − s U ( x , s ) = − 60 Solve the differential equation by the method undetermined coefficient. The auxiliary equation for the above differential equation as follows: r 2 − s = 0 Solve above auxiliary equation as follows: r 2 − s = 0 r 2 = s r = ± s ( Square root ) Therefore, the complementary function of the differential equation d 2 U d x 2 − s U ( x , s ) = − 60 is shown below. C .F = c 1 e − s x + c 2 e s x The particular integral of the differential equation d 2 U d x 2 − s U ( x , s ) = − 60 is obtained as follows: ( D 2 − s ) U ( x , s ) = − 60 P

Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604918

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.2, Problem 17E

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The solution of the heat equation uxx=ut using Laplace transform.

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Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

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 solution of the heat equation u x x = u t using Laplace transform.

Solution Summary: The author explains the Laplace transform for the solution of the heat equation u_xx.

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