Explanation Given: The given (non-self-adjoint) boundary value problem is, y ″ + 3 y ′ + 2 y = − x , y ( 0 ) = 0 , y ( 1 ) = 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: Consider the homogenous equation of the given problem is, y ″ + 3 y ′ + 2 y = 0 ... ( 1 ) The auxiliary equation of the ( 1 ) 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 = − 2 , − 1 . So, a general solution to the equation ( 1 ) is, y = A e − 2 x + B e − x . Suppose, z 1 ( x ) = A e − 2 x + B e − x . Apply the condition y ( 0 ) = 0 in z 1 ( x ) , this gives, A + B = 0 . This implies, A = − B . Suppose A = 1 then B = − 1 . Thus, z 1 ( x ) = e − 2 x − e − x Now, suppose z 2 ( x ) = y ( x ) = A e − x + B e − 2 x Apply the condition y ( 1 ) = 0 in z 2 ( x ) , this gives, A e − 2 + B e − 1 = 0 . This implies that, B = A e − 1 . As, A = 1 then B = e − 1 . Thus, z 2 ( x ) = e − x − e − 1 e − 2 x = e − x − e − 1 − 2 x Therefore, z 1 ( x ) = e − x − e − 2 x z 2 ( x ) = e − x − e 1 − 2 x 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 23E

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 22E

Chapter 11.6, Problem 24E

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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.