Explanation Given: The given (non-self-adjoint) boundary value problem is, x 2 y ″ − 2 x y ′ + 2 y = x , y ( 1 ) = 0 , y ( 2 ) = 0 . Approach: The solution of the given boundary value problem is given as, y ( x ) = ∫ a b K ( x , s ) f ( s ) d s , where K ( x , s ) = { − z 1 ( s ) z 2 ( x ) W [ z 1 , z 2 ] A 2 ( s ) , a ≤ s ≤ x , − z 1 ( x ) z 2 ( s ) W [ z 1 , z 2 ] A 2 ( s ) , x ≤ s ≤ b Here the function K ( x , s ) is called Green’s function. Calculation: Suppose, y = x k Then, y ′ = k x k − 1 y ″ = k ( k − 1 ) x k − 2 . Substitute the value of y , y ′ , and y ″ in the given problem, this gives, ( k 2 − 3 k + 2 ) x k = 0 . The auxiliary equation is given as, r 2 − 3 r + 2 = 0 . Thus, the roots of the above equation are, r 2 − 3 r + 2 = 0 r 2 − 2 r − r + 2 = 0 r ( r − 2 ) − ( r − 2 ) = 0 ( r − 1 ) ( r − 2 ) = 0 r = 1 , 2 . So, a general solution to the given homogeneous problem is, y = A x + B x 2 . Suppose, z 1 ( x ) = A x + B x 2 . Apply the condition y ( 1 ) = 0 in z 1 ( x ) , this gives, A + B = 0 . This implies, A = − B . Suppose A = 1 then B = − 1 . Thus, z 1 ( x ) = x − x 2 = x ( 1 − x ) Now, suppose z 2 ( x ) = y ( x ) = A x + B x 2 Apply the condition y ( 2 ) = 0 in z 2 ( x ) , this gives, 2 A + 4 B = 0 A = − 2 B Thus, z 2 ( x ) = − 2 x + x 2 = − x ( 2 − x ) Therefore, z 1 ( x ) = x − x 2 z 2 ( x ) = − 2 x + x 2 As Wronskian of z 1 ( x ) and z 2 ( x ) are given by, W [ z 1 , z 2 ] = z 1 z ′ 2 − z 2 z ′ 1 To determine (b) To find: The solution for the given (non-self-adjoint) boundary value problem.

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 11.6, Problem 25E

To determine

(a)

To find:

The Green’s function for the given (non-self-adjoint) boundary value problem.

To determine

(b)

To find:

The solution for the given (non-self-adjoint) boundary value problem.

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Chapter 11.6, Problem 24E

Chapter 11.6, Problem 26E

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(a) To find: The Green’s function for the given (non-self-adjoint) boundary value problem.

Solution Summary: The author explains the Green's function for the given (non-self-adjoint) boundary value problem.