Explanation Given: The given nonhomogenous boundary value problem is, [ ( 1 − x 2 ) y ′ ] ′ + μ y = f ( x ) , 0 < x < 1 , y ′ ( 0 ) = 0 , y ( x ) , y ′ ( x ) remain bounded as x → 1 − . Approach: For the given problem if, ∫ a b ( u L [ v ] − v L [ u ] ) ( x ) d x = lim x → b − p ( x ) W [ u , v ] ( x ) − lim x → a + p ( x ) W [ u , v ] ( x ) = 0 , Then corresponding homogenous problem of the given problem is selfadjoint. A general singular Sturm-Liouville boundary value problem associated with Legendre’s equation is given by, [ ( 1 − x 2 ) y ′ ] ′ + λ y = 0 , − 1 < x < 1 . y ( x ) , y ′ ( x ) remain bounded as x → ± 1 . The corresponding polynomial solutions of Sturm-Liouville boundary value problem are given as: P n ( x ) = 2 − n ∑ m = 0 [ n / 2 ] ( − 1 ) m ( 2 n − 2 m ) ! ( n − m ) ! m ! ( n − 2 m ) ! x n − 2 m , where [ n 2 ] is the greatest integer less than or equal to n 2 . A formal eifenfunction for this solution is: f ( x ) = ∑ n = 0 ∞ a n P n ( x ) . Here, a n = ∫ − 1 1 f ( x ) P n ( x ) d x ∫ − 1 1 P n 2 ( x ) d x . Calculation: Consider the nonhomogeneous boundary value problem is, [ ( 1 − x 2 ) y ′ ] ′ + μ y = f ( x ) . Thus, evaluate L [ v ] = [ ( 1 − x 2 ) v ′ ] ′ + μ v = ( 1 − x 2 ) v ″ − 2 x v ′ + μ v L [ u ] = [ ( 1 − x 2 ) u ′ ] ′ + μ u = ( 1 − x 2 ) u ″ − 2 x u ′ + μ 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,

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 6E

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

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Chapter 11.7, Problem 5E

Chapter 11.7, Problem 8E

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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 eigenfunction 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 eigenfunction 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 eigenfunction for the solution to the given nonhomogeneous boundary value problem.

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Chapter 11 Solutions

To verify:

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