Ch5 Complete Test Bank Series Solutions Of Second-Order - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.

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Ch5 Complete Test Bank Series Solutions Of Second-Order

Elementary Differential Equations, 12e (Boyce)

Chapter 5 Series Solutions of Second-Order Linear Equations

1) What is the radius of convergence of the power series sum of (((x - 8)) with superscript (n)) from (n = 0) to (∞)?

A) 0

B) 1

C) 8

D) ∞

Type: MC Var: 1

2) What is the radius of convergence of the power series sum of ((n/(4) with superscript (n))((x - 2)) with superscript (n)) from (n = 0) to (∞)?

A) (1/4)

B) 4

C) 2

D) 6

E) ∞

Type: MC Var: 1

3) What is the radius of convergence of the power series sum of ((((x - 6)) with superscript (2n)/7n)) from (n = 1) to (∞)?

A) 0

B) (1/6)

C) 1

D) 6

E) 7

Type: MC Var: 1

4) What is the radius of convergence of the power series sum of ((((x - 5)) with superscript (2n)/(36) with superscript (n))) from (n = 1) to (∞)?

A) 5

B) 6

C) 36

D) ∞

Type: MC Var: 1

5) What is the Taylor series expansion for f(x) = sin(6x) about x = 0?

A) sum of ((((-6)) with superscript (n)(x) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

B) sum of ((((-1)) with superscript (n)((6x)) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

C) sum of ((((-1)) with superscript (n)((6x)) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

D) sum of ((((-6)) with superscript (n)(x) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

Type: MC Var: 1

6) What is the Taylor expansion for f(x) = cos(7x) about x = 0?

A) sum of ((((-7)) with superscript (n)(x) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

B) sum of ((((-1)) with superscript (n)((7x)) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

C) sum of ((((-1)) with superscript (n)((7x)) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

D) sum of ((((-7)) with superscript (n)(x) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

Type: MC Var: 1

7) What is the Taylor series expansion for f(x) = (e) with superscript (-2x) about x = 0?

A) sum of ((((-2)) with superscript (n)(x) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

B) sum of ((((-1)) with superscript (n)((2x)) with superscript (2n + 1)/(2n + 1)!)) from (n = 0) to (∞)

C) sum of ((((-1)) with superscript (n)((2x)) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

D) sum of ((((-2)) with superscript (n)(x) with superscript (2n)/(2n)!)) from (n = 0) to (∞)

E) sum of ((((-1)) with superscript (n)((2x)) with superscript (n)/n!)) from (n = 0) to (∞)

Type: MC Var: 1

8) Which of these power series is equivalent to sum of ((n + 1)(n + 2)(a) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞)? Select all that apply.

A) sum of (n(n + 1)(a) with subscript (n - 1)(x) with superscript (n - 1)) from (n = 0) to (∞)

B) sum of ((n + 2)(n + 3)(a) with subscript (n + 1)(x) with superscript (n + 1)) from (n = 0) to (∞)

C) sum of ((n + 2)(n + 3)(a) with subscript (n + 1)(x) with superscript (n + 1)) from (n = 2) to (∞)

D) sum of (n(n + 1)(a) with subscript (n - 2)(x) with superscript (n - 2)) from (n = 3) to (∞)

E) sum of (n(n + 1)(a) with subscript (n - 1)(x) with superscript (n - 1)) from (n = 2) to (∞)

Type: MC Var: 1

9) Which of these power series is equivalent to sum of ((a) with subscript (n + 1)(x) with superscript (n)) from (n = 0) to (∞) + 7sum of ((a) with subscript (k + 2)(x) with superscript (k - 1)) from (k = 1) to (∞)?

A) sum of (((a) with subscript (n + 1) + 7(a) with subscript (n + 2))(x) with superscript (n)) from (n = 0) to (∞)

B) sum of (((a) with subscript (n) + 7(a) with subscript (n + 2))(x) with superscript (n)) from (n = 1) to (∞)

C) sum of (((a) with subscript (n) + 7(a) with subscript (n + 2))(x) with superscript (n)) from (n = 0) to (∞)

D) sum of (((a) with subscript (n) + 7(a) with subscript (n + 1))(x) with superscript (n - 1)) from (n = 1) to (∞)

E) sum of (((a) with subscript (n - 1) + 7(a) with subscript (n + 2))(x) with superscript (n + 1)) from (n = 0) to (∞)

Type: MC Var: 1

10) Which of these are singular points for the differential equation

(x) with superscript (2)y'' + (x - 6/x + 2)y' + (x - 2/((x + 3)) with superscript (2))y = 0?

Select all that apply.

A) -2

B) 6

C) -3

D) 2

E) 0

Type: MC Var: 1

11) Which of these are ordinary points for the differential equation

(x + 5)y'' + ((x) with superscript (2) - 49)y' + 2xy = 0?

Select all that apply.

A) -5

B) -7

C) 7

D) 0

E) 12

Type: MC Var: 1

12) Which of these are singular points for the differential equation

((x) with superscript (2) + 25)y'' + (x - 6/(x) with superscript (2) - 49)y' + (x/x - 8)y = 0?

Select all that apply.

A) -5

B) -7

C) 5

D) 6

E) 7

F) 8

Type: MC Var: 1

13) Consider the second-order differential equation y'' + 64y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

What is the recurrence relation for the coefficients (c) with subscript (n)? Assume that (c) with subscript (0) and (c) with subscript (1) are known.

A) (c) with subscript (n + 2) + 64(c) with subscript (n) = 0, n = 0, 1, 2, ...

B) (c) with subscript (n + 1) + 64(c) with subscript (n) = 0, n = 0, 1, 2, ...

C) (n + 1)(n + 2)(c) with subscript (n + 2) + 64(c) with subscript (n) = 0, n = 0, 1, 2, ...

D) n(n + 1)(c) with subscript (n + 1) + 64(c) with subscript (n) = 0, n = 0, 1, 2, ...

Type: MC Var: 1

14) Consider the second-order differential equation y'' + 49y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the explicit formulas for the coefficients (c) with subscript (n).

(c) with subscript (2n) = ________, n = 0, 1, 2, ...

(c) with subscript (2n + 1) = ________, n = 0, 1, 2, ...

(c) with subscript (2n+1) = ((-1)) with superscript (n)((7) with superscript (2n + 1)/(2n + 1)!)(c) with subscript (1), n = 0, 1, 2, ...

Type: ES Var: 1

15) Consider the second-order differential equation y'' + 100y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Assume the solution of the given differential equation is written as

y(x) = (c) with subscript (0) ∙ (y) with subscript (1)(x) + ((c) with subscript (1)/10) ∙ (y) with subscript (2)(x) = (c) with subscript (0)sum of ((c) with subscript (2n)(x) with superscript (2n)) from (n = 0) to (∞) + ((c) with subscript (1)/10)sum of (10 ∙ (c) with subscript (2n + 1)(x) with superscript (2n + 1)) from (n = 0) to (∞)

Identify elementary functions for (y) with subscript (1)(x) and (y) with subscript (2)(x).

(y) with subscript (1)(x) = ________

(y) with subscript (2)(x) = ________

(y) with subscript (2)(x) = cos(10x)

Type: ES Var: 1

16) Consider the second-order differential equation y'' - 4xy' + y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

What is the recurrence relation for the coefficients (c) with subscript (n)? Assume that (c) with subscript (0) and (c) with subscript (1) are known.

A) (c) with subscript (n + 2) - (4n - 1)(c) with subscript (n) = 0, n = 0, 1, 2, ...

B) (n + 1)(n + 2)(c) with subscript (n + 2) - (4n - 1)(c) with subscript (n) = 0, n = 0, 1, 2, ...

C) (n + 1)(c) with subscript (n + 1) - (4n - 1)(c) with subscript (n) = 0, n = 0, 1, 2, ...

D) (c) with subscript (n + 1) - (4n - 1)(c) with subscript (n) = 0, n = 0, 1, 2, ...

Type: MC Var: 1

17) Consider the second-order differential equation y'' - 2xy' + y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the following explicit formulas for the coefficients (c) with subscript (n):

(c) with subscript (2n) = ________, n = 0, 1, 2, ...

(c) with subscript (2n + 1) = ________, n = 0, 1, 2, ...

(c) with subscript (2n + 1) = ((1)(5)...((2n + 1)2 - 1)/(2n + 1)!)(c) with subscript (1), n = 0, 1, 2, ...

Type: ES Var: 1

18) Consider the first-order differential equation y' - 5y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

What is the recurrence relation for the coefficients (c) with subscript (n)? Assume that (c) with subscript (0) is known.

A) (c) with subscript (n + 1) - 5(c) with subscript (n) = 0, n = 0, 1, 2, ...

B) (n + 1)(n + 2)(c) with subscript (n + 1) - 5(c) with subscript (n) = 0, n = 0, 1, 2, ...

C) (n + 1)(c) with subscript (n + 1) + 5(c) with subscript (n) = 0, n = 0, 1, 2, ...

D) (n + 1)(c) with subscript (n + 1) - 5(c) with subscript (n) = 0, n = 0, 1, 2, ...

Type: MC Var: 1

19) Consider the first-order differential equation y' - 7y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the following explicit formula for the coefficients (c) with subscript (n):

(c) with subscript (n) = ________, n = 0, 1, 2, ...

Type: SA Var: 1

20) Consider the first-order differential equation y' - 2y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Which of these elementary functions is equal to the power series representation of the solution?

A) y = (2) with superscript ((c) with subscript (0))(e) with superscript (x)

B) y = (c) with subscript (0)(e) with superscript (2x)

C) y = (c) with subscript (0)(e) with superscript (x)

D) y = (c) with subscript (0)(e) with superscript ((x/2))

E) y = ((c) with subscript (0)/2)(e) with superscript (x)

Type: MC Var: 1

21) Consider the first-order differential equation y' - 17(x) with superscript (2)y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume that (c) with subscript (0) is known.

Which of these power series equals y(x)?

A) sum of ((17(c) with subscript (0)/3((n!)) with superscript (2))(x) with superscript (3n)) from (n = 0) to (∞)

B) sum of (((c) with subscript (0)/n!)((17/3)) with superscript (n)(x) with superscript (3n)) from (n = 0) to (∞)

C) sum of ((17(c) with subscript (0)/(3) with superscript (n)n!)(x) with superscript (3n)) from (n = 0) to (∞)

D) sum of ((17(c) with subscript (0)/(3) with superscript (n)((n!)) with superscript (2))(x) with superscript (3n)) from (n = 0) to (∞)

Type: MC Var: 1

22) Consider the first-order differential equation y' - 5(x) with superscript (2)y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume that (c) with subscript (0) is known.

Identify an elementary function equal to y(x).

Type: SA Var: 1

23) Consider this initial value problem: y'' - 5xy' - 5y = 0, y(0) = 1, y'(0) = 0

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the following explicit formulas for the coefficients (c) with subscript (n):

(c) with subscript (0) = ________,

(c) with subscript (1) = ________,

(c) with subscript (2n) = ________, n = 1, 2, ...

(c) with subscript (2n + 1) = ________, n = 1, 2, ...

(c) with subscript (1) = 0

(c) with subscript (2n) = ((5) with superscript (n)/(2) with superscript (n)n!), n = 1, 2, ...

(c) with subscript (2n + 1) = 0, n = 1, 2, ...

Type: ES Var: 1

24) Consider this initial value problem: y'' - 9xy' - 9y = 0, y(0) = 1, y'(0) = 0

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Express the solution y(x) as an elementary function.

A) y(x) = (e) with superscript ((((9x)) with superscript (2)/2))

B) y(x) = (e) with superscript ((9/2)(x) with superscript (2))

C) y(x) = 9(e) with superscript (((x/2)) with superscript (2))

D) y(x) = (9/2)(e) with superscript ((x) with superscript (2))

Type: MC Var: 1

25) Consider the first-order differential equation y' - 7xy = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

What is the recurrence relation for the coefficients (c) with subscript (n)? Assume that (c) with subscript (0) is known.

A) (c) with subscript (1) = 0, n(c) with subscript (n + 1) = 7(c) with subscript (n - 1), n = 1, 2, ...

B) (c) with subscript (0) = 0, (n + 1)(c) with subscript (n) = 7(c) with subscript (n - 1), n = 1, 2, ...

C) (c) with subscript (0) = 0, (n + 1)(c) with subscript (n + 1) = 7(c) with subscript (n - 1), n = 1, 2, ...

D) (c) with subscript (0) = 0, (n + 1)(c) with subscript (n + 1) = 7(c) with subscript (n), n = 0, 1, 2, ...

E) (c) with subscript (1) = 0, (n + 1)(c) with subscript (n + 1) = 7(c) with subscript (n - 1), n = 1, 2, ...

Type: MC Var: 1

26) Consider the first-order differential equation y' - 10xy = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Express the solution y(x) as an elementary function.

A) y(x) = (c) with subscript (0)(e) with superscript ((10/2)(x) with superscript (2))

B) y(x) = (c) with subscript (0)(e) with superscript (((10x)) with superscript (2))

C) y(x) = (c) with subscript (1)(e) with superscript (10x)

D) y(x) = (c) with subscript (1)(e) with superscript (10(x) with superscript (2))

E) y(x) = (c) with subscript (1)(e) with superscript (((10x)) with superscript (2))

Type: MC Var: 1

27) Consider the second-order differential equation y'' - 1(x) with superscript (2)y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Assume that (c) with subscript (0) and (c) with subscript (1) are known. Write down the following explicit formulas for the coefficients (c) with subscript (n):

(c) with subscript (2) = ________,

(c) with subscript (3) = ________,

(c) with subscript (n + 2) = ________, n = 2, 3, ...

(c) with subscript (3) = 0

(c) with subscript (n + 2) = (1(c) with subscript (n)/(n + 1)(n + 2))

Type: ES Var: 1

28) Consider the second-order differential equation y'' - 19(x) with superscript (2)y = 0.

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the first four nonzero terms of the power series solution.

y(x) ≈ ________

Type: SA Var: 1

29) Consider this initial-value problem: y'' + 5xy' - 15y = 0, y(0) = 9, y'(0) = 0

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the values of these coefficients:

(c) with subscript (0) = ________,

(c) with subscript (1) = ________,

(c) with subscript (2) = ________,

(c) with subscript (3) = ________,

(c) with subscript (4) = ________,

(c) with subscript (5) = ________,

(c) with subscript (6) = ________

(c) with subscript (1) = 0

(c) with subscript (2) = 1

(c) with subscript (3) = 0

(c) with subscript (4) = (5/12)

(c) with subscript (5) = 0

(c) with subscript (6) = (5/72)

Type: ES Var: 1

30) Consider this initial-value problem: y'' + 7xy' - 21y = 0, y(0) = 2, y'(0) = 0

Assume a solution of this equation can be represented as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞).

Write down the first four terms of the power series solution.

Type: SA Var: 1

31) Consider the second-order differential equation (x) with superscript (2)y'' + 3xy' + 32xy = 0. Assume the solution can be expressed as a power series y = sum of ((c) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (c) with subscript (0) = 0. Find (c) with subscript (1).

Type: SA Var: 1

32) What is the greatest lower bound for the radius of convergence of a series solution for the second-order differential equation (x + 8)y'' - 7xy' + 7y = 0 about the point (x) with subscript (0) = 14?

Type: SA Var: 1

33) What is the greatest lower bound for the radius of convergence of a series solution for the second-order differential equation (x - 2)(x + 4)y'' + 7(x + 16)y' - 8xy = 0 about the point (x) with subscript (0) = 0?

Type: SA Var: 1

34) What is a lower bound for the radius of convergence of a series solution for the second-order differential equation (x - 2)(x + 4)y'' + 8(x + 7)y' - 7xy = 0 about the point (x) with subscript (0) = -5?

Type: SA Var: 1

35) What is a lower bound for the radius of convergence of a series solution for the second-order differential equation (x - 6)(x + 12)y'' + 6(x + 14)y' - 2xy = 0 about the point (x) with subscript (0) = 11?

Type: SA Var: 1

36) What is the radius of convergence of a series solution for the second-order differential equation 4y'' + 2xy' + cos(π/3)xy = 0 about the point (x) with subscript (0) = 0?

A) 0

B) 1

C) (π/3)

D) ∞

Type: MC Var: 1

37) What is a lower bound for the radius of convergence of a series solution for the second-order differential equation (5(x) with superscript (2) + x + 5)y'' - 6y = 0 about the point (x) with subscript (0) = 15?

Type: SA Var: 1

38) What is the greatest lower bound of the radius of convergence of a series solution for the second-order differential equation (2(x) with superscript (2) + x + 2)y'' - 3y = 0 about the point (x) with subscript (0) = - (1/4)?

A) square root of (15)

B) (1/4)

C) (square root of (15)/4)

D) (1/2)

Type: MC Var: 1

39) Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation (x) with superscript (2)y'' - 8xy' + 18y = 0, x > 0?

A) {(x) with superscript (-3), (x) with superscript (3)}

B) {(x) with superscript (3), (x) with superscript (6)}

C) {(x) with superscript (3), (x) with superscript (3) ln x}

D) {(x) with superscript (-3), (x) with superscript (-3) ln x}

E) {(x) with superscript (3) cos(ln x), (x) with superscript (3) sin(ln x)}

Type: MC Var: 1

40) Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation (x) with superscript (2)y'' + 11xy' + 25y = 0, x > 0?

A) {(x) with superscript (-5), (x) with superscript (-5) cos(ln x)}

B) {(x) with superscript (5), (x) with superscript (-5)}

C) {(x) with superscript (5), (x) with superscript (5) ln x}

D) {(x) with superscript (-5), (x) with superscript (-5)ln x}

E) {(x) with superscript (5), (x) with superscript (5) cos(ln x)}

Type: MC Var: 1

41) Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation (x) with superscript (2)y'' + xy' + 36y = 0, x > 0?

A) {cos(6 ln x), sin(6 ln x)}

B) {(x) with superscript (6) cos(ln x), (x) with superscript (6) sin(ln x)}

C) {ln(cos(6x)), ln(sin(6x))}

D) {(x) with superscript (-6), (x) with superscript (6)}

E) {cos(ln(6x)), sin(ln(6x))}

Type: MC Var: 1

42) Find the general solution of the Cauchy Euler differential equation (x) with superscript (2)y'' + 2xy' + (1/2)y = 0, x > 0.

A) y = (C) with subscript (1)(x) with superscript (- (1/2)) + (C) with subscript (2)(x) with superscript ((1/2))

B) y = (C) with subscript (1)(x) with superscript ((1/2)) + (C) with subscript (2)(x) with superscript ((1/2))ln x

C) y = (C) with subscript (1)(x) with superscript (- (1/2)) + (C) with subscript (2)(x) with superscript (- (1/2))ln x

D) y = (x) with superscript (- (1/2))((C) with subscript (1) sin(1/2)ln x + (C) with subscript (2) cos(1/2)ln x)

E) y = (x) with superscript (- (1/2))((C) with subscript (1) sinln(1/2)x + (C) with subscript (2) cosln(1/2)x)

F) y = (x) with superscript ((1/2))((C) with subscript (1) sin- (1/2)ln x + (C) with subscript (2) cos- (1/2)ln x)

Type: MC Var: 1

43) Find the general solution of the Cauchy Euler differential equation 15(x) with superscript (2)y'' - 49xy' + 64y = 0, x > 0.

A) y = (C) with subscript (1)(x) with superscript (- (8/5)) + (C) with subscript (2)(x) with superscript (- (8/3))

B) y = (C) with subscript (1)(x) with superscript ((8/5)) + (C) with subscript (2)(x) with superscript ((8/3))

C) y = (C) with subscript (1)(x) with superscript ((5/8)) + (C) with subscript (2)(x) with superscript ((5/8)) ln x

D) y = (C) with subscript (1)(x) with superscript ((8/3)) + (C) with subscript (2)(x) with superscript ((8/3))ln x

E) y = (C) with subscript (1)(x) with superscript ((8/5)) + (C) with subscript (2)(x) with superscript (- (8/3))

Type: MC Var: 1

44) Find the general solution of the Cauchy Euler differential equation (x) with superscript (2)y'' + 6xy' + -14y = 0, x > 0.

A) y = (C) with subscript (1)(x) with superscript (-2) + (C) with subscript (2)(x) with superscript (-7)

B) y = (C) with subscript (1)(x) with superscript (2) + (C) with subscript (2)(x) with superscript (-7)

C) y = (C) with subscript (1)(x) with superscript (-2) + (C) with subscript (2)(x) with superscript (7)

D) y = (C) with subscript (1)(x) with superscript (2) + (C) with subscript (2)(x) with superscript (7)

Type: MC Var: 1

45) Solve this initial value problem:

(x) with superscript (2)y'' + -2xy' + -18y = 0, x > 0, y(1) = 5, y'(1) = 10

A) y = (25/9)(x) with superscript (6) + (25/9)(x) with superscript (-3)

B) y = (25/9)(x) with superscript (-6) + (25/9)(x) with superscript (3)

C) y = - (25/3)(x) with superscript (6) - (40/3)(x) with superscript (-3)

D) y = - (25/3)(x) with superscript (-6) - (40/3)(x) with superscript (3)

Type: MC Var: 1

46) Solve this initial value problem:

(x) with superscript (2)y'' + xy' + ((8π)) with superscript (2)y = 0, x > 0, y((e) with superscript (2)) = 2, y'((e) with superscript (2)) = (-4π/(e) with superscript (2))

Type: SA Var: 1

47) Find the general solution of the Cauchy Euler differential equation ((x - 3)) with superscript (2)y'' - 14(x - 3)y' + 50y = 0, x > 3.

A) y = (C) with subscript (1)((x - 3)) with superscript (5) + (C) with subscript (2)((x - 3)) with superscript (5)ln x

B) y = (C) with subscript (1)((x - 3)) with superscript (-5) + (C) with subscript (2)((x - 3)) with superscript (5)

C) y = (C) with subscript (1)((x - 3)) with superscript (-5) + (C) with subscript (2)((x - 3)) with superscript (-5) ln x

D) y = (C) with subscript (1)((x - 3)) with superscript (5) + (C) with subscript (2)((x - 3)) with superscript (10)

Type: MC Var: 1

48) Find the general solution of the Cauchy Euler differential equation ((x + 1)) with superscript (2)y'' + 15(x + 1)y' + 49y = 0, x > -1.

A) y = ((x + 1)) with superscript (-7)((C) with subscript (1) + (C) with subscript (2) ln x)

B) y = (C) with subscript (1)((x + 1)) with superscript (-7) + (C) with subscript (2)((x + 1)) with superscript (7)

C) y = ((x + 1)) with superscript (-7)((C) with subscript (1) sin(ln x) + (C) with subscript (2) cos(ln x))

D) y = ((x + 1)) with superscript (7)((C) with subscript (1) + (C) with subscript (2) ln x)

E) y = ((x + 1)) with superscript (7)((C) with subscript (1) ln(sin x)+ (C) with subscript (2) ln(cos x))

Type: MC Var: 1

49) Consider the Bessel equation of order 7: (x) with superscript (2)y'' + xy' + ((x) with superscript (2) - 49)y = 0.

Which of these statements is true?

A) x = 7 is a regular singular point.

B) x = 0 is a regular singular point.

C) x = 0 is an irregular singular point.

D) There are no singular points.

Type: MC Var: 1

50) Consider the Legendre equation: (1 - (x) with superscript (2))y'' - 2xy' + α(α + 1)y = 0.

Which of these statements is true?

A) x = 1 is a regular singular point and x = -1 is an irregular singular point.

B) x = 1 is an irregular singular point and x = -1 is a regular singular point.

C) Both x = 1 and x = -1 are regular singular points.

D) Both x = 1 and x = -1 are irregular singular points.

Type: MC Var: 1

51) Consider the second-order differential equation 3.2x((x - 5)) with superscript (2)y'' + 6.2xy' + (x - 5)y = 0.

Which of these statements is true?

A) x = 0 is a regular singular point and x = 5 is an irregular singular point.

B) x = 0 is an irregular singular point and x = 5 is a regular singular point.

C) Both x = 0 and x = 5 are regular singular points.

D) Both x = 0 and x = 5 are irregular singular points.

Type: MC Var: 1

52) Consider the second-order differential equation: (81 - (x) with superscript (2))y'' - 2xy' + 7y = 0.

Which of these statements is true?

A) x = 9 and x = -9 are both regular singular points.

B) x = 9 and x = -9 are both irregular singular points.

C) x = 0 and x = 9 are regular singular points, and x = -9 is an irregular singular point.

D) x = 0 and x = -9 are regular singular points, and x = 9 is an irregular singular point.

Type: MC Var: 1

53) Consider the second-order differential equation ((x - 8)) with superscript (3)y'' - 5y = 0.

Which of these statements is true?

A) x = 0 is a regular singular point.

B) x = -8 is a regular singular point.

C) x = -8 is an irregular singular point.

D) There are no singular points.

Type: MC Var: 1

54) x = 0 is a regular singular point for the second-order differential equation (x) with superscript (2)y'' + 4((e) with superscript (x) - 5)y' + ((e) with superscript (-7x) cos x)y = 0.

Type: TF Var: 1

55) Consider the second-order differential equation ((x - 16)) with superscript (2)y'' - 3xy' 4y = 0.

Which of these statements is true?

A) x = 4 and x = -4 are both irregular singular points.

B) x = 4 and x = -4 are both regular singular points.

C) x = -4 is a regular singular point and x = 4 is an irregular singular point.

D) x = 4 is a regular singular point and x = -4 is an irregular singular point.

Type: MC Var: 1

56) Consider the second-order differential equation: 5(x) with superscript (2)y'' + 7x(x + 1)y' - 7y = 0.

Why is (x) with subscript (0) = 0 a regular singular point?

A) The functions (x) with superscript (2) ∙ (7x(x + 1)/5(x) with superscript (2)) and x ∙ (- (7/5(x) with superscript (2))) both have convergent Taylor series expansions about 0.

B) The functions x ∙ (7x(x + 1)/5(x) with superscript (2)) and (x) with superscript (2) ∙ (- (7/5(x) with superscript (2))) both have convergent Taylor series expansions about 0.

C) (x → 0) is under (lim)x ∙ (- (7/5(x) with superscript (2))) = ∞

D) (x → 0) is under (lim)x ∙ (- (7/5(x) with superscript (2))) ≠ 0

Type: MC Var: 1

57) Consider the second-order differential equation: 3(x) with superscript (2)y'' + 5x(x + 1)y' - 5y = 0.

Which of these is the indicial equation?

A) (3r - 1)(r + 5) = 0

B) (3r + 1)(r - 5) = 0

C) (3r + 5)(r - 1) = 0

D) (3r - 5)(r + 1) = 0

Type: MC Var: 1

58) Consider the second-order differential equation: 2(x) with superscript (2)y'' + 5x(x + 1)y' - 5y = 0.

Which of these is the recurrence relation for the coefficients?

A) (a) with subscript (n) = (-2(n + r - 1)(a) with subscript (n - 1)/(2(n + r) + 5)(n + r + 1)), n ≥ 1

B) (a) with subscript (n) = (2(n + r - 1)(a) with subscript (n - 1)/(2(n + r) - 5)(n + r + 1)), n ≥ 1

C) (a) with subscript (n) = (-2(n + r - 1)(a) with subscript (n - 1)/(2(n + r) - 1)(n + r + 5)), n ≥ 1

D) (a) with subscript (n) = (2(n + r - 1)(a) with subscript (n - 1)/(2(n + r) + 1)(n + r - 5)), n ≥ 1

Type: MC Var: 1

59) Consider the second-order differential equation: 3(x) with superscript (2)y'' + 7x(x + 1)y' - 7y = 0.

Write out the first three terms of the solution corresponding to the positive root of the indicial equation.

(y) with subscript (1)(x) ≈ ________

Type: SA Var: 1

60) Consider the second-order differential equation: 7(x) with superscript (2)y'' + 11x(x + 1)y' - 11y = 0.

Write out the first three terms of the solution corresponding to the nonpositive root of the indicial equation.

(y) with subscript (2)(x) ≈ ________

Type: SA Var: 1

61) Consider the second-order differential equation: 7(x) with superscript (2)y'' + 10x(x + 1)y' - 10y = 0.

The general solution of the differential equation is y(x) = (C) with subscript (1)(y) with subscript (1)(x) + (C) with subscript (2)(y) with subscript (2)(x), where (C) with subscript (1) and (C) with subscript (2) are arbitrary real constants.

Type: TF Var: 1

62) Consider the Bessel equation of order 6: (x) with superscript (2)y'' + xy' + ((x) with superscript (2) - 36)y = 0

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Which of these is the indicial equation?

A) (r) with superscript (2) - 36 = 0

B) (r) with superscript (2) - 6 = 0

C) (r) with superscript (2) + 6 = 0

D) (r) with superscript (2) + 36 = 0

Type: MC Var: 1

63) Consider the Bessel equation of order 4: (x) with superscript (2)y'' + xy' + ((x) with superscript (2) - 16)y = 0

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Which of these is the recurrence relation for the coefficients?

A) (a) with subscript (1) = 0, (a) with subscript (n + 2) = (-(a) with subscript (n)/((r + n - 2)) with superscript (2) + 16), n ≥ 0

B) (a) with subscript (1) = 0, (a) with subscript (n + 2) = (-(a) with subscript (n)/((r + n + 2)) with superscript (2) - 16), n ≥ 0

C) (a) with subscript (1) = 1, (a) with subscript (n + 2) = (-(a) with subscript (n)/((r + n - 2)) with superscript (2) + 16), n ≥ 0

D) (a) with subscript (1) = 1, (a) with subscript (n + 2) = (-(a) with subscript (n)/((r + n + 2)) with superscript (2) - 16), n ≥ 0

Type: MC Var: 1

64) Consider the Bessel equation of order 4: (x) with superscript (2)y'' + xy' + ((x) with superscript (2) - 16)y = 0

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Which of these is the explicit formula for the coefficients corresponding to the positive root of the indicial equation?

A) (a) with subscript (2n) = 0 and (a) with subscript (2n + 1) = ((-1)) with superscript (n) ∙ ((a) with subscript (0) ∙ 4!/(2) with superscript (n + 1)(n + 1)!(n + 4)!), n ≥ 1

B) (a) with subscript (2n) = 0 and (a) with subscript (2n + 1) = ((-1)) with superscript (n) ∙ ((a) with subscript (0) ∙ 4!/(2) with superscript (2n)n!(n + 4)!), n ≥ 1

C) (a) with subscript (2n + 1) = 0 and (a) with subscript (2n) = ((-1)) with superscript (n - 1) ∙ ((a) with subscript (0) ∙ 4!/(2) with superscript (n)n!(n + 4)!), n ≥ 1

D) (a) with subscript (2n + 1) = 0 and (a) with subscript (2n) = ((-1)) with superscript (n) ∙ ((a) with subscript (0) ∙ 4!/(2) with superscript (2n)n!(n + 4)!), n ≥ 1

Type: MC Var: 1

65) Consider the Bessel equation of order 6: (x) with superscript (2)y'' + xy' + ((x) with superscript (2) - 36)y = 0

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Write the power series solution corresponding to the positive root of the indicial equation.

(y) with subscript (1)(x) = ________

Type: SA Var: 1

66) Consider the second-order differential equation x(1 - x)y'' + 8(1 - x)y' - 7y = 0

Write the differential equation in the form y'' + p(x)y' + q(x)y = 0. Why is (x) with subscript (0) = 0 a regular singular point for this equation?

A) (x) with superscript (2)p(x) and xq(x) both have convergent Taylor expansions about 0.

B) (x → 0) is under (lim)(x) with superscript (2)p(x) = 0 and (x → 0) is under (lim)xq(x) = ∞

C) (x → 0) is under (lim)xp(x) is finite and (x → 0) is under (lim)(x) with superscript (2)q(x) = ∞

D) (x → 0) is under (lim)xp(x) is finite and (x) with superscript (2)q(x) has a convergent Taylor expansion about 0.

Type: MC Var: 1

67) Consider the second-order differential equation x(1 - x)y'' + 8(1 - x)y' - 8y = 0

Suppose the method of Frobineius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0. Which of these is the indicial equation?

A) (r) with superscript (2) + 7r = 0

B) (r) with superscript (2) + 8r = 0

C) (r) with superscript (2) - 8r = 0

D) (r) with superscript (2) - 7r = 0

Type: MC Var: 1

68) Consider the second-order differential equation xy'' + 2y' - y = 0.

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Which of these is the indicial equation?

A) (r) with superscript (2) + r = 0

B) (r) with superscript (2) - r = 0

C) (r) with superscript (2) + r - 2 = 0

D) (r) with superscript (2) - r - 2 = 0

Type: MC Var: 1

69) Consider the second-order differential equation xy'' + 2y' - y = 0.

Suppose the method of Frobenius is used to determine a power series solution of the form y(x) = sum of ((a) with subscript (n)(x) with superscript (n + r)) from (n = 0) to (∞) of this differential equation. Assume (a) with subscript (0) ≠ 0.

Using the larger root of the indicial equation, write down an explicit formula for the coefficients and the corresponding power series solution.

(a) with subscript (n) = ________, n ≥ 1

(y) with subscript (1)(x) = ________

(y) with subscript (1)(x) = sum of (((a) with subscript (0)/n!(n + 1)!)(x) with superscript (n)) from (n = 0) to (∞)

Type: ES Var: 1

70) Consider the second-order differential equation xy'' + y' + 64xy = 0

Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series y(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (a) with subscript (0) ≠ 0.

Which of these is the recurrence relation for the coefficients?

A) (a) with subscript (1) = 0, (a) with subscript (n) = - (64 ∙ (a) with subscript (n - 2)/(2) with superscript (n)), n ≥ 2

B) (a) with subscript (1) = 1, (a) with subscript (n) = (64 ∙ (a) with subscript (n - 2)/(2) with superscript (n)), n ≥ 2

C) (a) with subscript (1) = 1, (a) with subscript (n) = (64 ∙ (a) with subscript (n - 2)/(n) with superscript (2)), n ≥ 2

D) (a) with subscript (1) = 0, (a) with subscript (n) = - (64 ∙ (a) with subscript (n - 2)/(n) with superscript (2)), n ≥ 2

E) (a) with subscript (1) = 0, (a) with subscript (n) = - (64 ∙ (a) with subscript (n - 2)/n), n ≥ 2

Type: MC Var: 1

71) Consider the second-order differential equation xy'' + y' + 32xy = 0

Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series y(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (a) with subscript (0) ≠ 0.

Which of these is the explicit formula for the coefficients (a) with subscript (2n)?

A) (a) with subscript (2n) = (((-1)) with superscript (n)(2) with superscript (3n)/n!)(a) with subscript (0), n ≥ 1

B) (a) with subscript (2n) = (((-1)) with superscript (n)(2) with superscript (5n)/n!)(a) with subscript (0), n ≥ 1

C) (a) with subscript (2n) = (((-1)) with superscript (n)(2) with superscript (3n)/((n!)) with superscript (2))(a) with subscript (0), n ≥ 1

D) (a) with subscript (2n) = (((-1)) with superscript (n)(2) with superscript (5n)/((n!)) with superscript (2))(a) with subscript (0), n ≥ 1

Type: MC Var: 1

72) Consider the second-order differential equation xy'' + y' + 64xy = 0

Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series y(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (a) with subscript (0) ≠ 0.

Assuming that (a) with subscript (0) = 1, one solution of the given differential equation is (y) with subscript (1)(x) = sum of ((a) with subscript (2n)(x) with superscript (2n)) from (n = 0) to (∞).

Assume x > 0. Which of these is a form of a second solution of the given differential equation, linearly independent to (y) with subscript (1)(x)?

A) (y) with subscript (2)(x) = (y) with subscript (1)(x)ln x + sum of (((a) with superscript (*)) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞)

B) (y) with subscript (2)(x) = (y) with subscript (1)(x)|ln x| + sum of (((a) with superscript (*)) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞)

C) (y) with subscript (2)(x) = ln x + (y) with subscript (1)(x) ∙ sum of (((a) with superscript (*)) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞)

D) (y) with subscript (2)(x) = |ln x| + (y) with subscript (1)(x) ∙ sum of (((a) with superscript (*)) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞)

E) (y) with subscript (2)(x) = ln x ((y) with subscript (1)(x) + sum of (((a) with superscript (*)) with subscript (n)(x) with superscript (n)) from (n = 1) to (∞))

Type: MC Var: 1

73) Consider the second-order differential equation xy'' + y' + 128xy = 0

Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series y(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (a) with subscript (0) ≠ 0.

Assuming that (a) with subscript (0) = 1, one solution of the given differential equation is (y) with subscript (1)(x) = sum of ((a) with subscript (2n)(x) with superscript (2n)) from (n = 0) to (∞).

Differentiating as needed, which of these relationships is correct?

A) 2y'(x) - ((a) with superscript (*)) with subscript (1) - 128((a) with superscript (*)) with subscript (2)x + sum of (((n) with superscript (2)((a) with superscript (*)) with subscript (n) - 128((a) with superscript (*)) with subscript (n - 2))(x) with superscript (n - 1)) from (n = 3) to (∞) = 0

B) 2y'(x) + ((a) with superscript (*)) with subscript (1) + 128((a) with superscript (*)) with subscript (2)x + sum of (((n) with superscript (2)((a) with superscript (*)) with subscript (n) + 128((a) with superscript (*)) with subscript (n - 2))(x) with superscript (n - 1)) from (n = 3) to (∞) = 0

C) 2y'(x) + ((a) with superscript (*)) with subscript (1) - 128((a) with superscript (*)) with subscript (2)x - sum of (((n) with superscript (2)((a) with superscript (*)) with subscript (n) + 128((a) with superscript (*)) with subscript (n - 2))(x) with superscript (n - 1)) from (n = 3) to (∞) = 0

D) 2y'(x) + ((a) with superscript (*)) with subscript (1) + 128((a) with superscript (*)) with subscript (2)x + sum of (((n) with superscript (2)((a) with superscript (*)) with subscript (n) - 128((a) with superscript (*)) with subscript (n - 2))(x) with superscript (n - 1)) from (n = 3) to (∞) = 0

Type: MC Var: 1

74) Consider the second-order differential equation xy'' + y' + 32xy = 0

Suppose the method of Frobenius is used to determine a power series solution of this equation. The indicial equation has r = 0 as a double root. So, one of the solutions can be represented as the power series y(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞). Assume (a) with subscript (0) ≠ 0.

Assuming that (a) with subscript (0) = 1, one solution of the given differential equation is (y) with subscript (1)(x) = sum of ((a) with subscript (2n)(x) with superscript (2n)) from (n = 0) to (∞).

Assuming that the coefficients ((a) with superscript (*)) with subscript (n) are known, what is the radius of convergence of the power series of the second solution (y) with subscript (2)(x)?

Type: SA Var: 1

75) Consider the second-order differential equation (x) with superscript (2)y'' + ((x) with superscript (2) - x)y' - (x - 1)y = 0, x > 0

Using the method of Frobenius, which of these is the general solution of this differential equation? Assume (a) with subscript (0) and ((a) with superscript (*)) with subscript (0) are arbitrary real constants.

A) y(x) = (a) with subscript (0) + ((a) with superscript (*)) with subscript (0)(x ln x + sum of ((((-1)) with superscript (n)(x) with superscript (n + 1)/n ∙ n!)) from (n = 1) to (∞))

B) y(x) = (a) with subscript (0)x + ((a) with superscript (*)) with subscript (0)(x ln x + sum of ((((-1)) with superscript (n)(x) with superscript (n + 1)/n ∙ n!)) from (n = 1) to (∞))

C) y(x) = (a) with subscript (0) + ((a) with superscript (*)) with subscript (0)x(ln x + sum of ((((-1)) with superscript (n)(x) with superscript (n + 1)/n ∙ n!)) from (n = 1) to (∞))

D) y(x) = (a) with subscript (0) + ((a) with superscript (*)) with subscript (0)ln x(1 + sum of ((((-1)) with superscript (n)(x) with superscript (n + 1)/n ∙ n!)) from (n = 1) to (∞))

Type: MC Var: 1

76) Consider the second-order differential equation (x) with superscript (2)y'' + ((x) with superscript (2) - x)y' - (x - 1)y = 0, x > 0

What is the radius of convergence of the series of the general solution of the differential equation?

A) 1

B) 2

C) 4

D) ∞

Type: MC Var: 1

77) Consider the second-order differential equation 7(x) with superscript (2)y'' + (x - (x) with superscript (2))y' - y = 0, x > 0

Suppose the method of Frobenius is used to determine the general solution of this differential equation.

Which of these is the indicial equation about the regular singular point x = 0?

A) (r) with superscript (2) + (6/7)r + (1/7) = 0

B) (r) with superscript (2) + (6/7)r - (1/7) = 0

C) (r) with superscript (2) - (1/7)r + (1/7) = 0

D) (r) with superscript (2) - (6/7)r - (1/7) = 0

Type: MC Var: 1

78) Consider the second-order differential equation 4(x) with superscript (2)y'' + (x - (x) with superscript (2))y' - y = 0, x > 0

Suppose the method of Frobenius is used to determine the general solution of this differential equation.

Which of the following is the form of a pair of linearly independent solution of this differential equation?

A) (y) with subscript (1)(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = (x) with superscript (- (1/4))sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

B) (y) with subscript (1)(x) = sum of ((a) with subscript (n)(x) with superscript (n + 1)) from (n = 0) to (∞), (y) with subscript (2)(x) = (x) with superscript (- (1/4))sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

C) (y) with subscript (1)(x) = sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = (x) with superscript ((3/4))sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

D) (y) with subscript (1)(x) = sum of ((a) with subscript (n)(x) with superscript (n + 1)) from (n = 0) to (∞), (y) with subscript (2)(x) = (x) with superscript ((3/4))sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

E) (y) with subscript (1)(x) = ln(x)sum of ((a) with subscript (n)(x) with superscript (n + 1)) from (n = 0) to (∞), (y) with subscript (2)(x) = (x) with superscript (- (1/4))sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

Type: MC Var: 1

79) Consider the second-order differential equation (x) with superscript (2)y'' - 7(x + (x) with superscript (2))y' + 16y = 0, x > 0

Which of these is the indicial equation about the regular singular point x = 0?

A) (r) with superscript (2) - 9r - 16 = 0

B) (r) with superscript (2) - 7r + 16 = 0

C) (r) with superscript (2) + 7r + 16 = 0

D) (r) with superscript (2) + 8r - 16 = 0

E) (r) with superscript (2) - 8r + 16 = 0

Type: MC Var: 1

80) Consider the second-order differential equation (x) with superscript (2)y'' - 7(x + (x) with superscript (2))y' + 16y = 0, x > 0

Which of the following is the form of a pair of linearly independent solutions of this differential equation?

A) (y) with subscript (1)(x) = (x) with superscript (-4)sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = ln x((y) with subscript (1)(x) + (x) with superscript (-4)sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞))

B) (y) with subscript (1)(x) = (x) with superscript (4)sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = ln x((y) with subscript (1)(x) + (x) with superscript (4)sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞))

C) (y) with subscript (1)(x) = (x) with superscript (-4)sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = (y) with subscript (1)(x)ln x + (x) with superscript (-4)sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

D) (y) with subscript (1)(x) = (x) with superscript (4)sum of ((a) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞), (y) with subscript (2)(x) = (y) with subscript (1)(x)ln x + (x) with superscript (4)sum of ((b) with subscript (n)(x) with superscript (n)) from (n = 0) to (∞)

Type: MC Var: 1

81) Consider the Bessel functions of the first kind of orders zero and one, respectively, given by

(J) with subscript (0) = 1 + sum of ((((-1)) with superscript (n)(x) with superscript (2n)/(2) with superscript (2n)((n!)) with superscript (2))) from (n = 1) to (∞)

(J) with subscript (1) = (x/2)sum of ((((-1)) with superscript (m)(x) with superscript (2m)/(2) with superscript (2m)(m + 1)!m!)) from (m = 0) to (∞)

Which of these are properties of these functions? Select all that apply.

A) (J) with subscript (0) has only finitely many zeroes for x > 0.

B) Both series converge absolutely for all real numbers x.

C) (J) with subscript (1)(x) = -(J) with subscript (0)'(x), for all real numbers x.

D) (J) with subscript (0)(x) → 0 as x → (0) with superscript (+)

E) (J) with subscript (0)(x) ≅ (((2/πx))) with superscript ((1/2))cos(x - (π/4)) as x → ∞

F) (J) with subscript (0)(x) ≅ (((2/πx))) with superscript ((1/2))sin(x - (π/4))

Type: MC Var: 1

© (2022) John Wiley & Sons, Inc. All rights reserved. Instructors who are authorized users of this course are permitted to download these materials and use them in connection with the course. Except as permitted herein or by law, no part of these materials should be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise.

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Document Type:
DOCX
Chapter Number:
5
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 5 Series Solutions Of Second-Order Linear Equations
Author:
William E. Boyce

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