Explanation Given: The given boundary value problem is: x 2 y ″ ( x ) + 2 x y ′ ( x ) + 5 4 y ( x ) = h ( x ) ; y ( 1 ) = 0 , y ( e π ) = 0 Procedure used: Let L be the differential operator given by the L [ y ] = A 2 y ″ + A 1 y ′ + A 0 y . The formal adjoint of L is the differential operator L + define by L + [ y ] = ( A 2 y ) ′′ − ( A 1 y ) ′ + A 0 y Provided A 2 ′′ , A 1 ′ and A 0 are continuous on [ a , b ] . If the roots of auxiliary equation having complex conjugate α ± β i . Then, The general solution is: y ( x ) = c 1 e α x cos β x + c 2 e α x sin β x Find the constant with the help of boundary conditions. Then, apply the Fredholm alternate theorem to determine the condition for solution. Fredholm alternate theorem: Let B is the set of linear boundary conditions. The nonhomogeneous boundary value problem: L [ y ] ( x ) = h ( x ) , a < x < b , B [ y ] = 0 has a solution if and only if ∫ a b h ( x ) z ( x ) d x = 0 Calculation: Consider the given linear differential operator L: x 2 y ″ ( x ) + 2 x y ′ ( x ) + 5 4 y ( x ) = h ( x ) Here A 0 = 5 4 , A 1 = 2 x and A 2 = x 2 . Substitute in the formal adjoint formula: L + [ y ] = ( x 2 ⋅ y ) ′′ − ( 2 x y ) ′ + 5 4 y = [ x 2 y ′ + 2 x y ] ′ − [ 2 x y ′ + 2 y ] + 5 4 y = [ x 2 y ″ + 2 x y ′ + 2 x y ′ + 2 y ] − 2 x y ′ − 2 y + 5 4 y = x 2 y ″ + 4 x y ′ + 2 y − 2 x y ′ − 3 4 y Further simplify as: x 2 y ″ + 4 x y ′ + 2 y − 2 x y ′ − 3 4 y = x 2 y ″ + 2 x y ′ + 5 4 y Let y = x r and substitute in the above equation: The auxiliary equation becomes: r 2 + r + 5 4 = 0 The roots of the equation are:

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

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

Publisher: PEARSON

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Chapter 11.4, Problem 20E

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The condition on h for the given nonhomogeneous boundary value problem.

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Chapter 11.4, Problem 19E

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Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

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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The condition on h for the given nonhomogeneous boundary value problem.

Solution Summary: The author explains the condition on h for the given nonhomogeneous boundary value problem. Let L be the differential operator given by Lleft[yright]=A_2y

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