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) ={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}ight. G(x,s) ={s,0≤x≤s−x,s≤x≤1​ 
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}ight. G(x,s) ={−x,0≤s≤xs,x≤s≤1​ 
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}ight. G(x,s) ={x,0≤x≤ss,s≤x≤1​ 
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}ight. G(x,s) ={s,0≤s≤xx,x≤s≤1​ 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

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) =∫01−G(−x,s) f(s) ds y(x) =\int_{0}^{1}-G(-x, s)  f(s)  d s y(x) =∫01​−G(−x,s) f(s) ds 
B)    y(x) =∫01−G(x,s) f(s) ds y(x) =\int_{0}^{1}-G(x, s)  f(s)  d s y(x) =∫01​−G(x,s) f(s) ds 
C)    y(x) =∫01G(−x,s) f(s) ds y(x) =\int_{0}^{1} G(-x, s)  f(s)  d s y(x) =∫01​G(−x,s) f(s) ds 
D)    y(x) =∫01G(x,s) f(s) ds y(x) =\int_{0}^{1} G(x, s)  f(s)  d s y(x) =∫01​G(x,s) f(s) ds 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)

Question 5

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

Accepted Answer

The answer of Consider the boundary value problem  
Determine...

Question 6

Consider the eigenfunction problem

What are the eigenvalues?

Accepted Answer

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)

Question 8

Determine the eigenfunctions for the eigenvalue problem

Accepted Answer

The answer of Determine the eigenfunctions for the eigenvalue problem
...

Question 9

Consider the eigenfunction problem
  
What are the corresponding eigenfunctions?

Accepted Answer

The answer of Consider the eigenfunction problem
  
What are...

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  λ\lambdaλ = -  satisfying the equation  .
B)   The positive eigenvalue  λ\lambdaλ  satisfies the equation  = -tan(6  ) .
C)    λ\lambdaλ  = 0 is an eigenvalue.
D)   There are no negative eigenvalues.
E)    λ\lambdaλ = 0 is not an eigenvalue.

Question 11

Consider the eigenfunction problem

Accepted Answer

The answer of Consider the eigenfunction problem
 ...

Question 12

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

Accepted Answer

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

Question 13

Consider the boundary value problem

Accepted Answer

The answer of Consider the boundary value problem
 ...

Question 14

Consider the boundary value problem

Accepted Answer

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)

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)    λ\lambdaλ  = 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
  
Evaluate...

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.