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Deck 11: Boundary Value Problems and Sturm-Liouville Theory

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Question
Consider the boundary value problem <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px>
Which of the following statements are true? Select all that apply.

A) λ\lambda = 0 is an eigenvalue.
B) There is one negative eigenvalue <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> such that tanh <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> (x) = C sinh( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> x), where C is an arbitrary nonzero real constant.
C) There are infinitely many positive eigenvalues <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> , n = 1, 2, 3, ... such that <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> (x) = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> sin( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> x), where <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> is an arbitrary nonzero real constant.
D) There are infinitely many negative eigenvalues <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> , n = 1, 2, 3, ... such that <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> (x) = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> sin( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> x), where <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. <div style=padding-top: 35px> is an arbitrary nonzero real constant.
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Question
Consider the boundary value problem
<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?

A)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
B)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
C)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
D)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
Question
Consider the boundary value problem
<strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. <div style=padding-top: 35px>
Which of the following statements are true? Select all that apply.

A) There are infinitely many negative eigenvalues λ\lambda = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. <div style=padding-top: 35px> satisfying the equation <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. <div style=padding-top: 35px> .
B) The positive eigenvalue λ\lambda satisfies the equation <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. <div style=padding-top: 35px> = -tan(6 <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. <div style=padding-top: 35px> ).
C) λ\lambda = 0 is an eigenvalue.
D) There are no negative eigenvalues.
E) λ\lambda = 0 is not an eigenvalue.
Question
Consider the Sturm-Liouville problem
<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>
Given the eigenfunctions of this boundary value problem are <strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px> .
Using this as an orthonormal basis, which of the following is the eigenfunction expansion of <strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>

A)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>
B)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>
C)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>
D)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) <div style=padding-top: 35px>
Question
Consider the Sturm-Liouville problem
Consider the Sturm-Liouville problem eigenfunction expansion of f(x) = 7x?<div style=padding-top: 35px>
eigenfunction expansion of f(x) = 7x?
Question
Consider the boundary value problem
Consider the boundary value problem This equation is in self-adjoint form.<div style=padding-top: 35px>
This equation is in self-adjoint form.
Question
Consider the boundary value problem
<strong>Consider the boundary value problem </strong> A) B) C) D) <div style=padding-top: 35px>

A) <strong>Consider the boundary value problem </strong> A) B) C) D) <div style=padding-top: 35px>
B) <strong>Consider the boundary value problem </strong> A) B) C) D) <div style=padding-top: 35px>
C) <strong>Consider the boundary value problem </strong> A) B) C) D) <div style=padding-top: 35px>
D) <strong>Consider the boundary value problem </strong> A) B) C) D) <div style=padding-top: 35px>
Question
Consider the boundary value problem
<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
What is the eigenfunction expansion of the solution y(x) of this boundary value problem?

A)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
B)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
C)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
D)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D) <div style=padding-top: 35px>
Question
Determine the eigenfunctions for the eigenvalue problem
Determine the eigenfunctions for the eigenvalue problem <div style=padding-top: 35px>
Question
Consider the eigenfunction problem
<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D) <div style=padding-top: 35px>
What are the eigenvalues?

A)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D) <div style=padding-top: 35px>
B)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D) <div style=padding-top: 35px>
C)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D) <div style=padding-top: 35px>
D)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D) <div style=padding-top: 35px>
Question
Consider the eigenfunction problem
Consider the eigenfunction problem What are the corresponding eigenfunctions?<div style=padding-top: 35px>
What are the corresponding eigenfunctions?
Question
Consider the eigenfunction problem
Consider the eigenfunction problem <div style=padding-top: 35px>
Question
Consider the boundary value problem
<strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px>
Which of these equations do the eigenvalues <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> satisfy?

A) sin(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> ) + <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> cos(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> ) = 0, n = 1, 2, 3, ...
B) sin(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> ) - <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> cos(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px> ) = 0, n = 1, 2, 3, ...
C) <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px>
D) <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) <div style=padding-top: 35px>
Question
Consider the boundary value problem Consider the boundary value problem Determine the normalized eigenfunctions (x).<div style=padding-top: 35px>
Determine the normalized eigenfunctions Consider the boundary value problem Determine the normalized eigenfunctions (x).<div style=padding-top: 35px> (x).
Question
Consider the boundary value problem
Consider the boundary value problem <div style=padding-top: 35px>
Question
Consider the boundary value problem
<strong>Consider the boundary value problem Which of these is the Green's function for this boundary value problem?</strong> A) G(x, s)=\left\{\begin{array}{l}s, 0 \leq x \leq s \\ -x, s \leq x \leq 1\end{array}\right. B) G(x, s)=\left\{\begin{array}{l}-x, 0 \leq s \leq x \\ s, x \leq s \leq 1\end{array}\right. C) G(x, s)=\left\{\begin{array}{l}x, 0 \leq x \leq s \\ s, s \leq x \leq 1\end{array}\right. D) G(x, s)=\left\{\begin{array}{l}s, 0 \leq s \leq x \\ x, x \leq s \leq 1\end{array}\right. <div style=padding-top: 35px>
Which of these is the Green's function for this boundary value problem?

A) G(x,s)={s,0≤x≤s−x,s≤x≤1 G(x, s)=\left\{\begin{array}{l}s, 0 \leq x \leq s \\ -x, s \leq x \leq 1\end{array}\right.
B) G(x,s)={−x,0≤s≤xs,x≤s≤1 G(x, s)=\left\{\begin{array}{l}-x, 0 \leq s \leq x \\ s, x \leq s \leq 1\end{array}\right.
C) G(x,s)={x,0≤x≤ss,s≤x≤1 G(x, s)=\left\{\begin{array}{l}x, 0 \leq x \leq s \\ s, s \leq x \leq 1\end{array}\right.
D) G(x,s)={s,0≤s≤xx,x≤s≤1 G(x, s)=\left\{\begin{array}{l}s, 0 \leq s \leq x \\ x, x \leq s \leq 1\end{array}\right.
Question
Consider the boundary value problem
- <strong>Consider the boundary value problem - = f(x), 0 < x < 1, y(0) = 0, (1) = 0 Which of these is the Green's function representation of the solution of the given boundary value problem?</strong> A) y(x)=\int_{0}^{1}-G(-x, s) f(s) d s B) y(x)=\int_{0}^{1}-G(x, s) f(s) d s C) y(x)=\int_{0}^{1} G(-x, s) f(s) d s D) y(x)=\int_{0}^{1} G(x, s) f(s) d s <div style=padding-top: 35px> = f(x), 0 < x < 1, y(0) = 0, <strong>Consider the boundary value problem - = f(x), 0 < x < 1, y(0) = 0, (1) = 0 Which of these is the Green's function representation of the solution of the given boundary value problem?</strong> A) y(x)=\int_{0}^{1}-G(-x, s) f(s) d s B) y(x)=\int_{0}^{1}-G(x, s) f(s) d s C) y(x)=\int_{0}^{1} G(-x, s) f(s) d s D) y(x)=\int_{0}^{1} G(x, s) f(s) d s <div style=padding-top: 35px> (1) = 0
Which of these is the Green's function representation of the solution of the given boundary value problem?

A) y(x)=∫01−G(−x,s)f(s)ds y(x)=\int_{0}^{1}-G(-x, s) f(s) d s
B) y(x)=∫01−G(x,s)f(s)ds y(x)=\int_{0}^{1}-G(x, s) f(s) d s
C) y(x)=∫01G(−x,s)f(s)ds y(x)=\int_{0}^{1} G(-x, s) f(s) d s
D) y(x)=∫01G(x,s)f(s)ds y(x)=\int_{0}^{1} G(x, s) f(s) d s
Question
Consider the boundary value problem
Consider the boundary value problem Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.<div style=padding-top: 35px>
Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.
Question
The singular Sturm-Liouville boundary value problem consisting of the differential equation The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint.<div style=padding-top: 35px> with boundary conditions that both y and The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint.<div style=padding-top: 35px> remain bounded as x approaches 0 from the right and that α\alpha y(1) + β\beta The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint.<div style=padding-top: 35px> (1) = 0 is self-adjoint.
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Deck 11: Boundary Value Problems and Sturm-Liouville Theory
1
Consider the boundary value problem <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant.
Which of the following statements are true? Select all that apply.

A) λ\lambda = 0 is an eigenvalue.
B) There is one negative eigenvalue <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. such that tanh <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. (x) = C sinh( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. x), where C is an arbitrary nonzero real constant.
C) There are infinitely many positive eigenvalues <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. , n = 1, 2, 3, ... such that <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. (x) = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. sin( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. x), where <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. is an arbitrary nonzero real constant.
D) There are infinitely many negative eigenvalues <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. , n = 1, 2, 3, ... such that <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. ; the corresponding eigenvectors are <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. (x) = <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. sin( <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. x), where <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) \lambda = 0 is an eigenvalue. B) There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. C) There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. D) There are infinitely many negative eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. is an arbitrary nonzero real constant.
There is one negative eigenvalue There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = - There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. such that tanh There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. ; the corresponding eigenvectors are There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. (x) = C sinh( There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. x), where C is an arbitrary nonzero real constant.
There are infinitely many positive eigenvalues There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. = - There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. , n = 1, 2, 3, ... such that There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. ; the corresponding eigenvectors are There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. (x) = There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. sin( There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. x), where There is one negative eigenvalue = - such that tanh = ; the corresponding eigenvectors are (x) = C sinh( x), where C is an arbitrary nonzero real constant. There are infinitely many positive eigenvalues = - , n = 1, 2, 3, ... such that ; the corresponding eigenvectors are (x) = sin( x), where is an arbitrary nonzero real constant. is an arbitrary nonzero real constant.
2
Consider the boundary value problem
<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D)
Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?

A)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D)
B)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D)
C)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D)
D)<strong>Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?</strong> A) B) C) D)

3
Consider the boundary value problem
<strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue.
Which of the following statements are true? Select all that apply.

A) There are infinitely many negative eigenvalues λ\lambda = - <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. satisfying the equation <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. .
B) The positive eigenvalue λ\lambda satisfies the equation <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. = -tan(6 <strong>Consider the boundary value problem Which of the following statements are true? Select all that apply.</strong> A) There are infinitely many negative eigenvalues \lambda = - satisfying the equation . B) The positive eigenvalue \lambda satisfies the equation = -tan(6 ). C) \lambda = 0 is an eigenvalue. D) There are no negative eigenvalues. E) \lambda = 0 is not an eigenvalue. ).
C) λ\lambda = 0 is an eigenvalue.
D) There are no negative eigenvalues.
E) λ\lambda = 0 is not an eigenvalue.
The positive eigenvalue λ\lambda satisfies the equation The positive eigenvalue \lambda satisfies the equation = -tan(6 ). There are no negative eigenvalues. \lambda = 0 is not an eigenvalue. = -tan(6 The positive eigenvalue \lambda satisfies the equation = -tan(6 ). There are no negative eigenvalues. \lambda = 0 is not an eigenvalue. ).
There are no negative eigenvalues.
λ\lambda = 0 is not an eigenvalue.
4
Consider the Sturm-Liouville problem
<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)
Given the eigenfunctions of this boundary value problem are <strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D) .
Using this as an orthonormal basis, which of the following is the eigenfunction expansion of <strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)

A)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)
B)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)
C)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)
D)<strong>Consider the Sturm-Liouville problem Given the eigenfunctions of this boundary value problem are . Using this as an orthonormal basis, which of the following is the eigenfunction expansion of </strong> A) B) C) D)
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5
Consider the Sturm-Liouville problem
Consider the Sturm-Liouville problem eigenfunction expansion of f(x) = 7x?
eigenfunction expansion of f(x) = 7x?
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6
Consider the boundary value problem
Consider the boundary value problem This equation is in self-adjoint form.
This equation is in self-adjoint form.
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7
Consider the boundary value problem
<strong>Consider the boundary value problem </strong> A) B) C) D)

A) <strong>Consider the boundary value problem </strong> A) B) C) D)
B) <strong>Consider the boundary value problem </strong> A) B) C) D)
C) <strong>Consider the boundary value problem </strong> A) B) C) D)
D) <strong>Consider the boundary value problem </strong> A) B) C) D)
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8
Consider the boundary value problem
<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D)
What is the eigenfunction expansion of the solution y(x) of this boundary value problem?

A)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D)
B)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D)
C)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D)
D)<strong>Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?</strong> A) B) C) D)
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9
Determine the eigenfunctions for the eigenvalue problem
Determine the eigenfunctions for the eigenvalue problem
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10
Consider the eigenfunction problem
<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D)
What are the eigenvalues?

A)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D)
B)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D)
C)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D)
D)<strong>Consider the eigenfunction problem What are the eigenvalues?</strong> A) B) C) D)
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11
Consider the eigenfunction problem
Consider the eigenfunction problem What are the corresponding eigenfunctions?
What are the corresponding eigenfunctions?
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12
Consider the eigenfunction problem
Consider the eigenfunction problem
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13
Consider the boundary value problem
<strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D)
Which of these equations do the eigenvalues <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) satisfy?

A) sin(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) ) + <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) cos(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) ) = 0, n = 1, 2, 3, ...
B) sin(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) ) - <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) cos(2 <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D) ) = 0, n = 1, 2, 3, ...
C) <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D)
D) <strong>Consider the boundary value problem Which of these equations do the eigenvalues satisfy?</strong> A) sin(2 ) + cos(2 ) = 0, n = 1, 2, 3, ... B) sin(2 ) - cos(2 ) = 0, n = 1, 2, 3, ... C) D)
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14
Consider the boundary value problem Consider the boundary value problem Determine the normalized eigenfunctions (x).
Determine the normalized eigenfunctions Consider the boundary value problem Determine the normalized eigenfunctions (x). (x).
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15
Consider the boundary value problem
Consider the boundary value problem
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16
Consider the boundary value problem
<strong>Consider the boundary value problem Which of these is the Green's function for this boundary value problem?</strong> A) G(x, s)=\left\{\begin{array}{l}s, 0 \leq x \leq s \\ -x, s \leq x \leq 1\end{array}\right. B) G(x, s)=\left\{\begin{array}{l}-x, 0 \leq s \leq x \\ s, x \leq s \leq 1\end{array}\right. C) G(x, s)=\left\{\begin{array}{l}x, 0 \leq x \leq s \\ s, s \leq x \leq 1\end{array}\right. D) G(x, s)=\left\{\begin{array}{l}s, 0 \leq s \leq x \\ x, x \leq s \leq 1\end{array}\right.
Which of these is the Green's function for this boundary value problem?

A) G(x,s)={s,0≤x≤s−x,s≤x≤1 G(x, s)=\left\{\begin{array}{l}s, 0 \leq x \leq s \\ -x, s \leq x \leq 1\end{array}\right.
B) G(x,s)={−x,0≤s≤xs,x≤s≤1 G(x, s)=\left\{\begin{array}{l}-x, 0 \leq s \leq x \\ s, x \leq s \leq 1\end{array}\right.
C) G(x,s)={x,0≤x≤ss,s≤x≤1 G(x, s)=\left\{\begin{array}{l}x, 0 \leq x \leq s \\ s, s \leq x \leq 1\end{array}\right.
D) G(x,s)={s,0≤s≤xx,x≤s≤1 G(x, s)=\left\{\begin{array}{l}s, 0 \leq s \leq x \\ x, x \leq s \leq 1\end{array}\right.
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17
Consider the boundary value problem
- <strong>Consider the boundary value problem - = f(x), 0 < x < 1, y(0) = 0, (1) = 0 Which of these is the Green's function representation of the solution of the given boundary value problem?</strong> A) y(x)=\int_{0}^{1}-G(-x, s) f(s) d s B) y(x)=\int_{0}^{1}-G(x, s) f(s) d s C) y(x)=\int_{0}^{1} G(-x, s) f(s) d s D) y(x)=\int_{0}^{1} G(x, s) f(s) d s = f(x), 0 < x < 1, y(0) = 0, <strong>Consider the boundary value problem - = f(x), 0 < x < 1, y(0) = 0, (1) = 0 Which of these is the Green's function representation of the solution of the given boundary value problem?</strong> A) y(x)=\int_{0}^{1}-G(-x, s) f(s) d s B) y(x)=\int_{0}^{1}-G(x, s) f(s) d s C) y(x)=\int_{0}^{1} G(-x, s) f(s) d s D) y(x)=\int_{0}^{1} G(x, s) f(s) d s (1) = 0
Which of these is the Green's function representation of the solution of the given boundary value problem?

A) y(x)=∫01−G(−x,s)f(s)ds y(x)=\int_{0}^{1}-G(-x, s) f(s) d s
B) y(x)=∫01−G(x,s)f(s)ds y(x)=\int_{0}^{1}-G(x, s) f(s) d s
C) y(x)=∫01G(−x,s)f(s)ds y(x)=\int_{0}^{1} G(-x, s) f(s) d s
D) y(x)=∫01G(x,s)f(s)ds y(x)=\int_{0}^{1} G(x, s) f(s) d s
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18
Consider the boundary value problem
Consider the boundary value problem Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.
Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.
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19
The singular Sturm-Liouville boundary value problem consisting of the differential equation The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint. with boundary conditions that both y and The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint. remain bounded as x approaches 0 from the right and that α\alpha y(1) + β\beta The singular Sturm-Liouville boundary value problem consisting of the differential equation with boundary conditions that both y and remain bounded as x approaches 0 from the right and that \alpha y(1) + \beta (1) = 0 is self-adjoint. (1) = 0 is self-adjoint.
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