Question 1

Consider the boundary value problem
 
Which of these is the Green's function for this boundary value problem?

Accepted Answer

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. $ 
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. $ C

Question 2

Consider the boundary value problem
 
Which of these equations do the eigenvalues   satisfy?

Accepted Answer

A)  sin(2   ) +   cos(2   ) = 0, n = 1, 2, 3, ... 
B)  sin(2   ) -   cos(2   ) = 0, n = 1, 2, 3, ... 
C)    
D)    
A)  sin(2   ) +   cos(2   ) = 0, n = 1, 2, 3, ... 
B)  sin(2   ) -   cos(2   ) = 0, n = 1, 2, 3, ... 
C)    
D)    A

Question 3

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?

Accepted Answer

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 $ 
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 $ D

Question 4

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

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 5

Consider the boundary value problem   
Determine the normalized eigenfunctions   (x).

Accepted Answer

@#IMG-DLM& (x) = @#IMG-DLM& ,

Question 6

Consider the eigenfunction problem
 
What are the eigenvalues?

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 7

Consider the boundary value problem
 
What is the eigenfunction expansion of the solution y(x) of this boundary value problem?

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 8

Determine the eigenfunctions for the eigenvalue problem

Accepted Answer

@#IMG-DLM& = C, where C is an

Question 9

Consider the eigenfunction problem
   
What are the corresponding eigenfunctions?

Accepted Answer

The answer of Consider the eigenfunction problem
   
What...

Question 10

Consider the boundary value problem
 
Which of the following statements are true? Select all that apply.

Accepted Answer

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. 
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.

Question 11

Consider the eigenfunction problem

Accepted Answer

@#IMG-DLM& = @#IMG-DLM& , n =

Question 12

Consider the Sturm-Liouville problem
   
eigenfunction expansion of f(x) = 7x?

Accepted Answer

The answer of Consider the Sturm-Liouville problem
   
eigenfunction...

Question 13

Consider the boundary value problem

Accepted Answer

@#IMG-DLM& = @#IMG-DLM& , n =

Question 14

Consider the boundary value problem

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 15

Consider the boundary value problem
 
Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 16

Consider the boundary value problem
   
This equation is in self-adjoint form.

Accepted Answer

A) True 
 B)False

Question 17

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.

Accepted Answer

A) True 
 B)False

Question 18

Consider the boundary value problem   
Which of the following statements are true? Select all that apply.

Accepted Answer

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. 
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.

Question 19

Consider the boundary value problem
   
Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.

Accepted Answer

The answer of Consider the boundary value problem
  ...

Consider the boundary value problem Which of these is the Green's function for this boundary value problem?

Consider the boundary value problem Which of these equations do the eigenvalues satisfy?

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?

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

Consider the boundary value problem Determine the normalized eigenfunctions (x).

Consider the eigenfunction problem What are the eigenvalues?

Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?

Determine the eigenfunctions for the eigenvalue problem

Consider the eigenfunction problem What are the corresponding eigenfunctions?

Consider the boundary value problem Which of the following statements are true? Select all that apply.

Consider the eigenfunction problem

Consider the Sturm-Liouville problem eigenfunction expansion of f(x) = 7x?

Consider the boundary value problem

Consider the boundary value problem

Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?

Consider the boundary value problem This equation is in self-adjoint form.

Consider the boundary value problem Which of the following statements are true? Select all that apply.

Consider the boundary value problem Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Filters

Exam 11: Boundary Value Problems and Sturm-Liouville Theory

Consider the boundary value problem Which of these is the Green's function for this boundary value problem?

Consider the boundary value problem Which of these equations do the eigenvalues satisfy?

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?

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

Consider the boundary value problem Determine the normalized eigenfunctions (x).

Consider the eigenfunction problem What are the eigenvalues?

Consider the boundary value problem What is the eigenfunction expansion of the solution y(x) of this boundary value problem?

Determine the eigenfunctions for the eigenvalue problem

Consider the eigenfunction problem What are the corresponding eigenfunctions?

Consider the boundary value problem Which of the following statements are true? Select all that apply.

Consider the eigenfunction problem

Consider the Sturm-Liouville problem eigenfunction expansion of f(x) = 7x?

Consider the boundary value problem

Consider the boundary value problem

Consider the boundary value problem Which of the following is a complete list of the eigenvalue-eigenvector pairs for this boundary value problem?

Consider the boundary value problem This equation is in self-adjoint form.

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 α\alphaα y(1) + β\betaβ (1) = 0 is self-adjoint.

Consider the boundary value problem Which of the following statements are true? Select all that apply.

Consider the boundary value problem Evaluate the Green's function representation of the solution when f(x) = 17x + 9, 0 ≤ x ≤ 1.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Filters