Explanation Given: The given (non-self-adjoint) boundary value problem is, y ″ − 2 y ′ + y = 1 , 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 ″ − 2 y ′ + y = 0 ... ( 1 ) The auxiliary equation of the ( 1 ) is given as, r 2 − 2 r + 1 = 0 . Thus, the roots of the above equation are, r 2 − 2 r + 1 = 0 r 2 − r − r + 1 = 0 r ( r − 1 ) − ( r − 1 ) = 0 ( r − 1 ) ( r − 1 ) = 0 r = 1 , 1 . So, a general solution to the equation ( 1 ) is, y = ( A + B x ) e x . Suppose, z 1 ( x ) = ( A + B x e ) x . Apply the condition y ( 0 ) = 0 in z 1 ( x ) , this gives, A = 0 . Suppose B = 1 . Thus, z 1 ( x ) = x e x Now, suppose z 2 ( x ) = y ( x ) = ( A + B x ) e x Apply the condition y ( 1 ) = 0 in z 2 ( x ) , this gives, 0 + B e = 0 . Thus, z 2 ( x ) = ( 1 − x ) e x Therefore, z 1 ( x ) = x e x z 2 ( x ) = ( 1 − x ) e 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

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 11.6, Problem 24E

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.6, Problem 23E

Chapter 11.6, Problem 25E

Students have asked these similar questions

Show that the Laplace equation in Cartesian coordinates: J²u J²u + = 0 მx2 Jy2 can be reduced to the following form in cylindrical polar coordinates: 湯( ди 1 8²u + Or 7,2 მ)2 = 0.

Draw the following graph on the interval πT 5π < x < x≤ 2 2 y = 2 cos(3(x-77)) +3 6+ 5 4- 3 2 1 /2 -π/3 -π/6 Clear All Draw: /6 π/3 π/2 2/3 5/6 x 7/6 4/3 3/2 5/311/6 2 13/67/3 5 Question Help: Video Submit Question Jump to Answer

Determine the moment about the origin O of the force F4i-3j+5k that acts at a Point A. Assume that the position vector of A is (a) r =2i+3j-4k, (b) r=-8i+6j-10k, (c) r=8i-6j+5k

Chapter 11 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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,...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

(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.