Ch11 Test Questions & Answers + Boundary Value Problems And - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.

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Ch11 Test Questions & Answers + Boundary Value Problems And

Elementary Differential Equations, 12e (Boyce)

Chapter 11 Boundary Value Problems and Sturm-Liouville Theory

1) Consider the boundary value problem

y'' + λy = 0, 0 < x < (π/1), y(0) = 0, 8y((π/1)) - y'((π/1)) = 0

Which of the following statements are true? Select all that apply.

A) λ = 0 is an eigenvalue.

B) There is one negative eigenvalue (λ) with subscript (0) = -((μ) with subscript (0)) with superscript (2) such that tanh((μ) with subscript (0) ∙ (π/1)) = ((μ) with subscript (0)/8); the corresponding eigenvectors are (y) with subscript (0)(x) = C sinh((μ) with subscript (0)x), where C is an arbitrary nonzero real constant.

C) There are infinitely many positive eigenvalues (λ) with subscript (n) = -((μ) with subscript (n)) with superscript (2), n = 1, 2, 3, ... such that tan(μ) with subscript (n) ∙ (π/1) = ((μ) with subscript (n)/8); the corresponding eigenvectors are (y) with subscript (n)(x) = (C) with subscript (n) sin((μ) with subscript (n)x), where (C) with subscript (n) is an arbitrary nonzero real constant.

D) There are infinitely many negative eigenvalues (λ) with subscript (n) = -((ν) with subscript (n)) with superscript (2), n = 1, 2, 3, ... such that tanh(ν) with subscript (n) ∙ (π/1) = ((ν) with subscript (n)/8); the corresponding eigenvectors are (y) with subscript (n)(x) = (C) with subscript (n) sin((ν) with subscript (n)x), where (C) with subscript (n) is an arbitrary nonzero real constant.

Type: MC Var: 1

2) Consider the boundary value problem

y'' + λy = 0, 0 < x < (π/6), y(0) = 0, y((π/6)) = 0

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

A) λ = 0, y = C, where C is an arbitrary nonzero real constant.

B) λ = 36(n) with superscript (2), n = 1, 2, 3, ... ; (y) with subscript (n) = (C) with subscript (n)sin(6nx), where (C) with subscript (n) is an arbitrary nonzero real constant.

C) λ = 6(n) with superscript (2), n = 1, 2, 3, ... ; (y) with subscript (n) = (C) with subscript (n)sin(6nx), where (C) with subscript (n) is an arbitrary nonzero real constant.

D) λ = -36(n) with superscript (2), n = 1, 2, 3, ... ; (y) with subscript (n) = (C) with subscript (n)(e) with superscript (-6nx), where (C) with subscript (n) is an arbitrary nonzero real constant.

Type: MC Var: 1

3) Consider the boundary value problem

y'' + λy = 0, 0 < x < 6, y(0) = 0, y'(6) + y(6) = 0

Which of the following statements are true? Select all that apply.

A) There are infinitely many negative eigenvalues λ = -(μ) with superscript (2) satisfying the equation μ = tanh(6μ).

B) The positive eigenvalue λ satisfies the equation square root of (λ) = -tan(6square root of (λ)).

C) λ = 0 is an eigenvalue.

D) There are no negative eigenvalues.

E) λ = 0 is not an eigenvalue.

Type: MC Var: 1

4) Consider the Sturm-Liouville problem

y'' + λy = 0, 0 < x < 5, y(0) = y(5) = 0

Given the eigenfunctions of this boundary value problem are {square root of ((2/5))sin(nπx/5) : n = 1, 2, 3, ...}. Using this as an orthonormal basis, which of the following is the eigenfunction expansion of f(x) = 2?

A) square root of ((2/5))sum of ((b) with subscript (n) sin(nπx/5)) from (n = 1) to (∞), where (b) with subscript (n) = {table ( (0, n is even)((4square root of (5)/nπ), n is odd) )

B) square root of ((2/5))sum of ((b) with subscript (n) sin(nπx/5)) from (n = 1) to (∞), where (b) with subscript (n) = {table ( ((4square root of (5)/nπ), n is even)(0, n is odd) )

C) square root of ((2/5))sum of ((b) with subscript (n) sin(nπx/5)) from (n = 1) to (∞), where (b) with subscript (n) = {table ( (0, n is even)((2square root of (5)/nπ), n is odd) )

D) square root of ((2/5))sum of ((b) with subscript (n) sin(nπx/5)) from (n = 1) to (∞), where (b) with subscript (n) = {table ( ((2square root of (5)/nπ), n is even)(0, n is odd) )

Type: MC Var: 1

5) Consider the Sturm-Liouville problem

y'' + λy = 0, 0 < x < 7, y(0) = y(7) = 0

Assume the eigenfunctions of this boundary value problem are

{square root of ((1/7))} ∪ {square root of ((2/7))cos(nπx/7) : n = 1, 2, 3, ...}. Using this as an orthonormal basis, what is the eigenfunction expansion of f(x) = 7x?

Type: SA Var: 1

6) Consider the boundary value problem

(xy')' + (y/x) = (1/x), 1 ≤ x ≤ (e) with superscript (3), y(1) = 0 = y((e) with superscript (3))

This equation is in self-adjoint form.

Type: TF Var: 1

7) Consider the boundary value problem

(xy')' + (y/x) = (1/x), 1 ≤ x ≤ (e) with superscript (5), y(1) = 0 = y((e) with superscript (5))

Which of these is an orthogonal set of eigenfunctions for the associated Sturm-Liouville problem (xφ')' + (φ/x) = -λrφ, φ(1) = 0 = φ((e) with superscript (5)), where r is a suitable weight function?

(Hint: Use r = (1/x).)

A) {(φ) with subscript (n)(x) = square root of ((2/5))sin(nπx) : n = 1, 2, 3, ... }

B) {(φ) with subscript (n)(x) = square root of ((5/2))sin(nπ ln x) : n = 1, 2, 3, ... }

C) {(φ) with subscript (n)(x) = square root of ((2/5))sin(nπ ln x) : n = 1, 2, 3, ... }

D) {(φ) with subscript (n)(x) = square root of ((5/2))sin(nπx) : n = 1, 2, 3, ... }

Type: MC Var: 1

8) Consider the boundary value problem

(xy')' + (y/x) = (1/x), 1 ≤ x ≤ (e) with superscript (3), y(1) = 0 = y((e) with superscript (3))

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

A) y(x) = sum of ((2[((-1)) with superscript (n) - 1]/3nπ((n) with superscript (2)(π) with superscript (2) - 1))sin(nπx)) from (n = 1) to (∞)

B) y(x) = sum of ((2[((-1)) with superscript (n) - 1]/3nπ((n) with superscript (2)(π) with superscript (2) - 1))sin(nπ ln x)) from (n = 1) to (∞)

C) y(x) = sum of ((square root of (2)[((-1)) with superscript (n) - 1]/3nπ((n) with superscript (2)(π) with superscript (2) - 1))sin(nπ ln x)) from (n = 1) to (∞)

D) y(x) = sum of ((square root of (2)[((-1)) with superscript (n) - 1]/3nπ((n) with superscript (2)(π) with superscript (2) - 1))sin(nπx)) from (n = 1) to (∞)

Type: MC Var: 1

9) Determine the eigenfunctions for the eigenvalue problem

(xy')' + (9/x)y = -λ(x) with superscript (-1)y, y'(1) = 0, y'(9) = 0

(y) with subscript (n) = cos((nπ/ln 9)ln x), 1 ≤ x ≤ 9, n = 1, 2, 3, ...

Type: SA Var: 1

10) Consider the eigenfunction problem

y'' - (1/4)y' + λy = 0, 0 < x < 25, y(0) = 0 = y'(25)

What are the eigenvalues?

A) (λ) with subscript (n) = (1/64) + (μ) with subscript (n), where (μ) with subscript (n) > 0 satisfies the equation tan(25square root of ((μ) with subscript (n))) = 8square root of ((μ) with subscript (n)), n = 1, 2, 3, ...

B) (λ) with subscript (n) = (1/64) - (μ) with subscript (n), where (μ) with subscript (n) > 0 satisfies the equation tan(25square root of ((μ) with subscript (n))) = 8square root of ((μ) with subscript (n)), n = 1, 2, 3, ...

C) (λ) with subscript (n) = - (1/64) + (μ) with subscript (n), where (μ) with subscript (n) > 0 satisfies the equation tan(25square root of ((μ) with subscript (n))) = -8square root of ((μ) with subscript (n)), n = 1, 2, 3, ...

D) (λ) with subscript (n) = (1/64) + (μ) with subscript (n), where (μ) with subscript (n) > 0 satisfies the equation tan(25square root of ((μ) with subscript (n))) = -8square root of ((μ) with subscript (n)), n = 1, 2, 3, ...

Type: MC Var: 1

11) Consider the eigenfunction problem

y'' - (1/3)y' + λy = 0, 0 < x < 50, y(0) = 0 = y'(50)

What are the corresponding eigenfunctions?

Type: SA Var: 1

12) Consider the eigenfunction problem

y'' - (1/3)y' + λy = 0, 0 < x < 35, y(0) = 0 = y'(35)

Find a formula for the constants (A) with subscript (n) for the eigenfunction expansion of f(x) = 9x + 5; that is, 9x + 5 = sum of ((A) with subscript (n)(e) with superscript ((x/6))(d) with subscript (n) sin(square root of ((μ) with subscript (n)) x)) from (n = 1) to (∞). Your formula will involve (μ) with subscript (n). Do not compute the integrals.

Type: SA Var: 1

13) Consider the boundary value problem

y'' + λy = 0, 0 < x < 2, y'(0) = y(2) + y'(2) = 0

Which of these equations do the eigenvalues (λ) with subscript (n) satisfy?

A) sin(2square root of ((λ) with subscript (n))) + square root of ((λ) with subscript (n)) cos(2square root of ((λ) with subscript (n))) = 0, n = 1, 2, 3, ...

B) sin(2square root of ((λ) with subscript (n))) - square root of ((λ) with subscript (n)) cos(2square root of ((λ) with subscript (n))) = 0, n = 1, 2, 3, ...

C) cos(2square root of ((λ) with subscript (n))) + square root of ((λ) with subscript (n)) sin(2square root of ((λ) with subscript (n))) = 0, n = 1, 2, 3, ...

D) cos(2square root of ((λ) with subscript (n))) - square root of ((λ) with subscript (n)) sin(2square root of ((λ) with subscript (n))) = 0, n = 1, 2, 3, ...

Type: MC Var: 1

14) Consider the boundary value problem

y'' + λy = 0, 0 < x < 5, y'(0) = y(5) + y'(5) = 0

Determine the normalized eigenfunctions (φ) with subscript (n)(x).

Type: SA Var: 1

15) Consider the boundary value problem

y'' + λy = 0, 0 < x < 3, y'(0) = y(3) + y'(3) = 0

Find a formula for the constants (c) with subscript (n) of the eigenfunction expansion sum of ((c) with subscript (n)(φ) with subscript (n)(x)) from (n = 1) to (∞) of f(x) = 7x, 0 ≤ x ≤ 3 using the normalized eigenfunctions (φ) with subscript (n)(x).

Type: SA Var: 1

16) Consider the boundary value problem

-y'' = f(x), 0 < x < 1, y(0) = 0, y'(1) = 0

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

A) G(x, s) = {table ( (s, 0 ≤ x ≤ s)(-x, s ≤ x ≤ 1) )

B) G(x, s) = {table ( (-x, 0 ≤ s ≤ x)(s, x ≤ s ≤ 1) )

C) G(x, s) = {table ( (x, 0 ≤ x ≤ s)(s, s ≤ x ≤ 1) )

D) G(x, s) = {table ( (s, 0 ≤ s ≤ x)(x, x ≤ s ≤ 1) )

Type: MC Var: 1

17) Consider the boundary value problem

-y'' = f(x), 0 < x < 1, y(0) = 0, y'(1) = 0

Which of these is the Green's function representation of the solution of the given boundary value problem?

A) y(x) = integral of (-G(-x, s) f(s) ds) from (0) to (1)

B) y(x) = integral of (-G(x, s) f(s) ds) from (0) to (1)

C) y(x) = integral of (G(-x, s) f(s) ds) from (0) to (1)

D) y(x) = integral of (G(x, s) f(s) ds) from (0) to (1)

Type: MC Var: 1

18) Consider the boundary value problem

-y'' = f(x), 0 < x < 1, y(0) = 0, y'(1) = 0

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

Type: SA Var: 1

19) The singular Sturm-Liouville boundary value problem consisting of the differential equation (3x + 2)y'' + y' + λxy = 0 with boundary conditions that both y and y' remain bounded as x approaches 0 from the right and that αy(1) + βy'(1) = 0 is self-adjoint.

Type: TF Var: 1

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Document Type:
DOCX
Chapter Number:
11
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 11 Boundary Value Problems And Sturm-Liouville Theory
Author:
William E. Boyce

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