Explanation Given: The boundary conditions are as follows: ∂ u ∂ x | x = 1 = 100 − u ( 1 , t ) or u x ( 1 , t ) = 100 − u ( 1 , t ) lim x → − ∞ u ( x , t ) = 0 u ( x , 0 ) = 0 Solution: Take Laplace transform on both sides of equation ∂ 2 u ∂ x 2 = ∂ 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 x ( 1 , t ) = 100 − u ( 1 , t ) as follows: L { u x ( 1 , t ) } = L { 100 − u ( 1 , t ) } U x ( 1 , s ) = 100 s − U ( 1 , s ) ...... ( 2 ) Take Laplace transform of boundary condition lim x → − ∞ u ( x , t ) = 0 as follows: L { lim x → − ∞ u ( x , t ) } = L { 0 } lim x → − ∞ U ( x , s ) = 0 ...... ( 3 ) Substitute u ( x , 0 ) = 0 in equation (1). d 2 U d x 2 = s U ( x , s ) d 2 U d x 2 − s U ( x , s ) = 0 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 ) = 0 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 ) = 0 is obtained as follows: d 2 U d x 2 − s U ( x , s ) = 0 ( D 2 − s ) U ( x , s ) = 0 P .I = 0 D 2 − s ( P

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Chapter 14.2, Problem 19E

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The solution of boundary value problem ∂2u∂x2=∂u∂t using Laplace transform.

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Chapter 14.2, Problem 18E

Chapter 14.2, Problem 20E

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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 boundary value problem ∂ 2 u ∂ x 2 = ∂ u ∂ t using Laplace transform.

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Chapter 14 Solutions