Explanation Given: The given nonhomogenous boundary value problem is, ( x y ′ ) ′ + μ x y = f ( x ) , 0 < x < 1 , y ( x ) , y ′ ( x ) remain bounded as x → 0 + , y ′ ( 1 ) = 0 . Approach: For the given problem if, ∫ a b ( u L [ v ] − v L [ u ] ) ( x ) d x = 0 , then corresponding homogenous problem of the given problem is selfadjoint. The corresponding Eigen functions are given by, y n ( x ) = ∑ n = 1 ∞ a n J v ( α v n x ) , where a n = ∫ 0 1 f ( x ) J v ( α v n x ) x d x ∫ 0 1 J v 2 ( α v n x ) x d x , n = 1 , 2 , 3 , ... Calculation: Consider the nonhomogeneous boundary value problem as, ( x y ′ ) + μ x y = f ( x ) . Thus, evaluate L [ v ] = ( x v ′ ) ′ + μ x v = x v ″ + v ′ + μ x v L [ u ] = ( x u ′ ) ′ + μ x u = x u ″ + u ′ + μ x u By the formula, ∫ a b ( u L [ v ] − v L [ u ] ) ( x ) d x = 0 . Evaluate the Left hand side of this equation as follows, ∫ a b ( u L [ v ] − v L [ u ] ) ( x ) d x = [ ∫ 0 1 u [ x v ″ + v ′ + μ x v ] − v [ x u ″ + u ′ + μ x u ] ] ( x ) d x = ∫ 0 1 [ x u v ″ + u v ′ + μ x u v − x u ″ v − u ′ v − μ x u v ] ( x ) d x = ∫ 0 1 x [ u ( x ) v ″ ( x ) + u ′ ( x ) v ′ ( x ) − u ′ ( x ) v ′ ( x ) − u ″ ( x ) v ( x ) ] d x = ∫ 0 1 x [ u ( x ) v ″ ( x ) + u ′ ( x ) v ′ ( x ) − ( u ′ ( x ) v ′ ( x ) + u ″ ( x ) v ( x ) ) ] d x As, u ( x ) v ″ ( x ) + u ′ ( x ) v ′ ( x ) = ( u ( x ) v ′ ( x ) ) ′ , and, u ′ ( x ) v ′ ( x ) + u ″ ( x ) v ( x ) = ( u ′ ( x ) v ( x ) ) ′ , Thus, the above equation becomes, ∫ a b ( u L [ v ] − v L [ u ] ) ( x ) d x = ∫ 0 1 x [ ( u ( x ) v ′ ( x ) ) ′ − ( u ′ ( x ) v ( x ) ) ′ ] d x = [ [ x ∫ ( u ( x ) v ′ ( x ) ) ′ d x − ∫ ( u ( x ) v ′ ( x ) ) d x ] 0 1 − [ x ∫ ( u ′ ( x ) v ′ ( x ) ) ′ d x − ∫ ( u ′ ( x ) v ( x ) ) d x ] 0 1 ] = [ [ x ( u ( x ) v ′ ( x ) ) − u ( x ) v ( x ) + ∫ ( u ′ ( x ) v ( x ) ) d x ] 0 1 − [ x ( u ′ ( x ) v ( x ) ) − v ( x ) u ( x ) + ∫ ( v ′ ( x ) u ( x ) ) d x ] 0 1 ] As, it is given that, y ′ ( 1 ) = 0 , this implies that, u ( 1 ) = 0 , and v ′ ( 1 ) = 0

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 11.7, Problem 3E

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The corresponding homogenous problem of the given problem is selfadjoint and find a formal Eigen function for the solution to the given nonhomogeneous boundary value problem.

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Chapter 11.7, Problem 2E

Chapter 11.7, Problem 4E

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Chapter 11 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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To verify: The corresponding homogenous problem of the given problem is selfadjoint and find a formal Eigen function for the solution to the given nonhomogeneous boundary value problem.

Solution Summary: The author explains that the corresponding homogenous problem is selfadjoint and finds a formal Eigen function for the solution to the given nonhomogeneous boundary value problem.

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To verify: The corresponding homogenous problem of the given problem is selfadjoint and find a formal Eigen function for the solution to the given nonhomogeneous boundary value problem.

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To verify:

The corresponding homogenous problem of the given problem is selfadjoint and find a formal Eigen function for the solution to the given nonhomogeneous boundary value problem.