Singular SturmLiouville Problemswhere f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem. Hint: Use a series expansion similar to those in Section 11.3.
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Table of Contents
1.1
Some Basic Mathematical Models; Direction Fields
1.2
Solutions of Some Differential Equations
1.3
Classification of Differential Equations
2
First Order Difference Equations
2.1
Linear Equations; Method of Integrating Factors
2.2
Separable Equations
2.3
Modeling with First Order Equations
2.4
Differences Between Linear and Nonlinear Equations
2.5
Autonomous Equations and Population Dynamics
2.6
Exact Equations and Integrating Factors
2.7
Numerical Approximations: Eulers Method
2.8
The Existence and Uniqueness Theorem
2.9
First Order Difference Equations
3.1
Homogeneous Equations with Constant Coefficients
3.2
Solutions of Linear Homogeneous Equations; the Wronskian
3.3
Complex Roots of the Characteristic Equation
3.4
Repeated Roots; Reduction of Order
3.5
Nonhomogeneous Equations; Method of Undetermined Coefficients
3.6
Variation of Parameters
3.7
Mechanical and Electrical Vibrations
3.8
Forced Vibrations
4.1
General Theory of nth Order Linear Equations
4.2
Homogeneous Equations with Constant Coefficients
4.3
The Method of Undetermined Coefficients
4.4
The Method of Variation of Parameters
5.1
Review of Power Series
5.2
Series Solutions Near an Ordinary Point, Part I
5.3
Series Solutions Near an Ordinary Point, Part II
5.4
Euler Equations; Regular Singular Points
5.5
Series Solutions Near a Regular Singular Point, Part I
5.6
Series Solutions Near a Regular Singular Point, Part II
5.7
Bessels Equation
6.1
Definition of the Laplace Transform
6.2
Solution of Initial Value Problems
6.3
Step Functions
6.4
Differential Equations with Discontinuous Forcing Functions
6.5
Impulse Functions
6.6
The Convolution Integral
7.1
Introduction
7.2
Review of Matrices
7.3
Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
7.4
Basic Theory of Systems of First Order Linear Equations
7.5
Homogeneous Linear Systems with Constant Coefficients
7.6
Complex Eigenvalues
7.7
Fundamental Matrices
7.8
Repeated Eigenvalues
7.9
Nonhomogeneous Linear Systems
8.1
The Euler or Tangent Line Method
8.2
Improvements on the Euler Method
8.3
The RungeKutta Method
8.4
Multistep Methods
8.5
More on Errors; Stability
8.6
Systems of First Order Equations
9.1
The Phase Plane: Linear Systems
9.2
Autonomous Systems and Stability
9.3
Locally Linear Systems
9.4
Competing Species
9.5
PredatorPrey Equations
9.6
Liapunovs Second Method
9.7
Periodic Solutions and Limit Cycles
9.8
Chaos and Strange Attractors: The Lorenz Equations
10.1
Two-Point Boundary Value Problems
10.2
Fourier Series
10.3
The Fourier Convergence Theorem
10.4
Even and Odd Functions
10.5
Separation of Variables; Heat Conduction in a Rod
10.6
Other Heat Conduction Problems
10.7
The Wave Equation: Vibrations of an Elastic String
10.8
Laplaces Equation
11.1
The Occurrence of Two-Point Boundary Value Problems
11.2
SturmLiouville Boundary Value Problems
11.3
Nonhomogeneous Boundary Value Problems
11.4
Singular SturmLiouville Problems
11.5
Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
11.6
Series of Orthogonal Functions: Mean Convergence
Textbook Solutions for Elementary Differential Equations and Boundary Value Problems
Chapter 11.4 Problem 2
Question
Consider the boundary value problem (xy ) = xy, y, y bounded as x 0, y (1) = 0. (a) Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 0(x) = 1. If > 0, show formally that the eigenfunctions are given by n(x) = J0( n x), where n is the nth positive root (in increasing order) of the equation J 0( ) = 0. It is possible to show that there is an infinite sequence of such roots. (b) Show that if m, n = 0, 1, 2, ... , then 1 0 xm(x)n(x) dx = 0, m = n. (c) Find a formal solution to the nonhomogeneous problem (xy ) = xy + f(x), y, y bounded as x 0, y (1) = 0, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.
Solution
The first step in solving 11.4 problem number 2 trying to solve the problem we have to refer to the textbook question: Consider the boundary value problem (xy ) = xy, y, y bounded as x 0, y (1) = 0. (a) Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 0(x) = 1. If > 0, show formally that the eigenfunctions are given by n(x) = J0( n x), where n is the nth positive root (in increasing order) of the equation J 0( ) = 0. It is possible to show that there is an infinite sequence of such roots. (b) Show that if m, n = 0, 1, 2, ... , then 1 0 xm(x)n(x) dx = 0, m = n. (c) Find a formal solution to the nonhomogeneous problem (xy ) = xy + f(x), y, y bounded as x 0, y (1) = 0, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.
From the textbook chapter Singular SturmLiouville Problems you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Elementary Differential Equations and Boundary Value Problems 9
Author
Boyce, Richard C. DiPrima
ISBN
9780470383346
Solved: Consider the boundary value problem (xy ) = xy, y, y bounded as x 0, y (1) = 0
Chapter 11.4 textbook questions
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Chapter 11: Problem 1 Elementary Differential Equations and Boundary Value Problems 9
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Chapter 11: Problem 2 Elementary Differential Equations and Boundary Value Problems 9
Consider the boundary value problem (xy ) = xy, y, y bounded as x 0, y (1) = 0. (a) Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 0(x) = 1. If > 0, show formally that the eigenfunctions are given by n(x) = J0( n x), where n is the nth positive root (in increasing order) of the equation J 0( ) = 0. It is possible to show that there is an infinite sequence of such roots. (b) Show that if m, n = 0, 1, 2, ... , then 1 0 xm(x)n(x) dx = 0, m = n. (c) Find a formal solution to the nonhomogeneous problem (xy ) = xy + f(x), y, y bounded as x 0, y (1) = 0, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.
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Chapter 11: Problem 3 Elementary Differential Equations and Boundary Value Problems 9
Consider the boundary value problem (xy ) = xy, y, y bounded as x 0, y (1) = 0. (a) Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 0(x) = 1. If > 0, show formally that the eigenfunctions are given by n(x) = J0( n x), where n is the nth positive root (in increasing order) of the equation J 0( ) = 0. It is possible to show that there is an infinite sequence of such roots. (b) Show that if m, n = 0, 1, 2, ... , then 1 0 xm(x)n(x) dx = 0, m = n. (c) Find a formal solution to the nonhomogeneous problem (xy ) = xy + f(x), y, y bounded as x 0, y (1) = 0, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.
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Chapter 11: Problem 4 Elementary Differential Equations and Boundary Value Problems 9
Consider Legendres equation (see Problems 22 through 24 of Section 5.3) [(1 x2 )y ] = y subject to the boundary conditions y(0) = 0, y, y bounded as x 1. The eigenfunctions of this problem are the odd Legendre polynomials 1(x) = P1(x) = x, 2(x) = P3(x) = (5x3 3x)/2, ... , n(x) = P2n1(x), ... corresponding to the eigenvalues 1 = 2, 2 = 4 3, ... , n = 2n(2n 1), .... (a) Show that 1 0 m(x)n(x) dx = 0, m = n. (b) Find a formal solution of the nonhomogeneous problem [(1 x2 )y ] = y + f(x), y(0) = 0, y, y bounded as x 1, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.
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Chapter 11: Problem 5 Elementary Differential Equations and Boundary Value Problems 9
The equation (1 x2 )y xy + y = 0 (i) is Chebyshevs equation; see Problem 10 of Section 5.3. (a) Show that Eq. (i) can be written in the form [(1 x2 ) 1/2 y ] = (1 x2 ) 1/2 y, 1 < x < 1. (ii) (b) Consider the boundary conditions y, y bounded as x 1, y, y bounded as x 1. (iii) Show that the boundary value problem (ii), (iii) is self-adjoint. (c) It can be shown that the boundary value problem (ii), (iii) has the eigenvalues 0 = 0, 1 = 1, 2 = 4, ... , n = n2, .... The corresponding eigenfunctions are the Chebyshev polynomials Tn(x): T0(x) = 1, T1(x) = x, T2(x) = 1 2x2, .... Show that 1 1 Tm(x)Tn(x) (1 x2)1/2 dx = 0, m = n. (iv) Note that this is a convergent improper integral.
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