Explanation It is given that ∂ u ∂ t = ∂ 2 u ∂ x 2 , 0 < x < π , t > 0 , (1) u ( 0 , t ) = 0 , u ( π , t ) + ∂ u ∂ x ( π , t ) = 0 , t > 0 , (2) u ( x , 0 ) = f ( x ) , 0 < x < π (3) Consider the solution of the equation be u ( x , t ) = X ( x ) T ( t ) . (4) Above equation satisfies equation (1), so it becomes, X ″ ( x ) + λ X ( x ) = 0 (5) And T ′ ( t ) + k λ T ( t ) = 0 (6) Boundary condition in equation (2) becomes, X ( 0 ) T ( t ) = 0 And X ( π ) T ( t ) = − X ′ ( π ) T ( t ) Either T ( t ) = 0 or X ( 0 ) = 0 , X ( π ) = − X ′ ( π ) So, equation (5) becomes, X ″ ( x ) + λ X ( x ) = 0 with boundary conditions X ( 0 ) = 0 and X ( π ) = − X ′ ( π ) . (7) The auxiliary equation becomes r 2 + λ = 0 . If λ = 0 , then r 2 = 0 r = 0 Thus r = 0 are the repeated roots. So, the general solution becomes X ( x ) = C 1 + C 2 x . By boundary condition (7), X ( 0 ) = C 1 0 = C 1 C 1 = 0 And X ( π ) = C 1 + C 2 π X ( π ) = C 2 ( π ) X ′ ( π ) = C 2 Since X ( π ) = − X ′ ( π ) , then, C 2 π = C 2 C 2 ( 1 − π ) = 0 C 2 = 0 As, C 1 = 0 and C 2 = 0 , then X ( x ) = 0 which means there is a no non trivial solution. Now, if λ > 0 , then the roots becomes, r 2 + λ = 0 r 2 = − λ r = ± i λ The general solution becomes, X ( x ) = c 1 cos λ x + c 2 sin λ x By boundary condition (7), X ( 0 ) = c 1 cos 0 + c 2 sin 0 0 = c 1 c 1 = 0 And X ( π ) = c 1 cos λ π + c 2 sin λ π X ( π ) = 0 X ′ ( π ) = − λ c 1 sin λ π + λ c 2 cos λ π X ′ ( π ) = λ c 2 cos λ π Since, X ( π ) = − X ′ ( π ) , then, λ c 2 cos λ π = 0 Here λ ≠ 0 , c 2 ≠ 0 , then

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 10.5, Problem 12E

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The formal solution to the given initial value problem.

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Chapter 10 Solutions

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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Solution Summary: The author explains that the formal solution to the given initial value problem is u(x,t)=a_02+displaystyle underset

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To find: The formal solution to the given initial value problem.

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Chapter 10 Solutions

To find:

The formal solution to the given initial value problem.