Deck 5: Series Solutions of Second-Order Linear Equations

Question 1

What is the radius of convergence of the power series ?

A) 0
B) 1
C) 8
D) $\infty$

Answer 1

1

Question 2

What is the radius of convergence of the power series ?

A)
B) 4
C) 2
D) 6
E) $\infty$

Answer 2

4

Question 3

What is the radius of convergence of the power series ?

A) 0
B)
C) 1
D) 6
E) 7

Answer 3

1

Question 4

What is the radius of convergence of the power series ?

A) 5
B) 6
C) 36
D) $\infty$

Question 5

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

A)
B)
C)
D)

Question 6

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

A)
B)
C)
D)

Question 7

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

A)
B)
C)
D)
E)

Question 8

Which of these power series is equivalent to ? Select all that apply.

A)
B)
C)
D)
E)

Question 9

Which of these power series is equivalent to + 7 ?

A)
B)
C) ?
D) ?
E)

Question 10

Which of these are singular points for the differential equation

Select all that apply.

A) -2
B) 6
C) -3
D) 2
E) 0

Question 11

Which of these are ordinary points for the differential equation
(x + 5) + ( - 49) + 2xy = 0?
Select all that apply.

A) -5
B) -7
C) 7
D) 0
E) 12

Question 12

Which of these are singular points for the differential equation

Select all that apply.

A) -5
B) -7
C) 5
D) 6
E) 7
F) 8

Question 13

Consider the second-order differential equation + 64y = 0.
Assume a solution of this equation can be represented as a power series
What is the recurrence relation for the coefficientsC_n ? Assume that C₀ and C₁ are known.

A) $c_{n+2}+64 c_{n}=0, n=0,1,2, \ldots$
B) $c_{n+1}+64 c_{n}=0, n=0,1,2, \ldots$
C) $(n+1)(n+2) c_{n+2}+64 c_{n}=0, n=0,1,2, \ldots$
D) $n(n+1) c_{n+1}+64 c_{n}=0, n=0,1,2, \ldots$

Question 14

Consider the second-order differential equation + 49y = 0.
Assume a solution of this equation can be represented as a power series
Write down the explicit formulas for the coefficients C_n

Question 15

Consider the second-order differential equation + 100y = 0.
Assume a solution of this equation can be represented as a power series
Assume the solution of the given differential equation is written as

Identify elementary functions for y₁ (x) and y₂ (x).
y₁ (x) = ________
y₂ (x) = ________

Question 16

Consider the second-order differential equation - 4x + y = 0.
Assume a solution of this equation can be represented as a power series
What is the recurrence relation for the coefficients C_n? Assume that C₀ and C₁ are known

A) $c_{n+2}-(4 n-1) c_{n}=0, n=0,1,2, \ldots$
B) $(n+1)(n+2) c_{n+2}-(4 n-1) c_{n}=0, n=0,1,2, \ldots$
C) $(n+1) c_{n+1}-(4 n-1) c_{n}=0, n=0,1,2, \ldots$
D) $c_{n+1}-(4 n-1) c_{n}=0, n=0,1,2, \ldots$

Question 17

Consider the second-order differential equation
Assume a solution of this equation can be represented as a power series
Write down the following explicit formulas for the coefficients C_n:
C_2n= ________, n = 0, 1, 2, ...
C_2n+1= ________, n = 0, 1, 2, ...

Question 18

Consider the first-order differential equation - 5y = 0.
Assume a solution of this equation can be represented as a power series
What is the recurrence relation for the coefficients C_n? Assume that C₀ is known

A) $c_{n+1}-5 c_{n}=0, n=0,1,2, \ldots$
B) $(n+1)(n+2) c_{n+1}-5 c_{n}=0, n=0,1,2, \ldots$
C) $(n+1) c_{n+1}+5 c_{n}=0, n=0,1,2, \ldots$
D) $(n+1) c_{n+1}-5 c_{n}=0, n=0,1,2, \ldots$

Question 19

Consider the first-order differential equation - 7y = 0.
Assume a solution of this equation can be represented as a power series
Write down the following explicit formula for the coefficients C_n
= , n = 0, 1, 2, ...

Question 20

Consider the first-order differential equation - 7y = 0.
Assume a solution of this equation can be represented as a power series
Which of these elementary functions is equal to the power series representation of the solution?

A) $y=2^{c_{0}} e^{x}$
B) $y=c_{0} e^{2 x}$
C) $y=c_{0} e^{x}$
D) $y=c_{0} e^{\frac{x}{2}}$
E) $y=\frac{c_{0}}{2} e^{x}$

Question 21

Consider the first-order differential equation
Assume a solution of this equation can be represented as a power series .
Assume that C₀ is known.
Which of these power series equals y(x)?

A) $\sum_{n=0}^{\infty} \frac{17 c_{0}}{3(n !)^{2}} x^{3 n}$
B) $\sum_{n=0}^{\infty} \frac{c_{0} 17^{n}}{n ! 3} x^{3 n}$
C) $\sum_{n=0}^{\infty} \frac{17 c_{0}}{3^{n} n !} x^{3 n}$
D) $\sum_{n=0}^{\infty} \frac{17 c_{0}}{3^{n}(n !)^{2}} x^{3 n}$

Question 22

Consider the first-order differential equation
Assume a solution of this equation can be represented as a power series
Assume that C₀ is known.
Identify an elementary function equal to y(x).

Question 23

Consider the first-order differential equation
.
Assume a solution of this equation can be represented as a power series .
Write down the following explicit formulas for the coefficients C_n
.

Question 24

Consider this initial value problem: .
Assume a solution of this equation can be represented as a power series .
Express the solution y(x) as an elementary function.

A) $y(x)=e^{\frac{(9 x)^{2}}{2}}$
B) $y(x)=e^{\frac{9}{2} x^{2}}$
C) $y(x)=9 e^{\frac{x^{2}}{2}}$
D) $y(x)=\frac{9}{2} e^{x^{2}}$

Question 25

Consider the first-order differential equation . - 7xy = 0.
Assume a solution of this equation can be represented as a power series .
What is the recurrence relation for the coefficients C_n ? Assume that C₀ is known.

A) $c_{1}=0, n c_{n+1}=7 c_{n-1}, n=1,2, \ldots$
B) $c_{0}=0,(n+1) c_{n}=7 c_{n-1}, n=1,2, \ldots$
C) $c_{0}=0,(n+1) c_{n+1}=7 c_{n-1}, n=1,2, \ldots$
D) $c_{0}=0,(n+1) c_{n+1}=7 c_{n}, n=0,1,2, \ldots$
E) $c_{1}=0,(n+1) c_{n+1}=7 c_{n-1}, n=1,2, \ldots$

Question 26

Consider the first-order differential equation - 10xy = 0.
Assume a solution of this equation can be represented as a power series
Express the solution y(x) as an elementary function.

A) $y(x)=c_{0} e^{\frac{10}{2} x^{2}}$
B) $y(x)=c_{0} e^{(10 x)^{2}}$
C) $y(x)=c_{1} e^{10 x}$
D) $y(x)=c_{1} e^{10 x^{2}}$
E) $y(x)=c_{1} e^{(10 x)^{2}}$

Question 27

Consider the second-order differential equation - 1 y = 0.
Assume a solution of this equation can be represented as a power series
Assume that C₀ and C₁ are known. Write down the following explicit formulas for the coefficients C_n
.

Question 28

Consider the second-order differential equation ‪ - 19x² y = 0.
Assume a solution of this equation can be represented as a power series
Write down the first four nonzero terms of the power series solution.
y(x) ≈ ________

Question 29

Consider this initial-value problem: .
Assume a solution of this equation can be represented as a power series .
Write down the values of these coefficients:
C₀ = ________,
C₁ = ________,
C₂ = ________,
C₃ = ________,
C₄ = ________,
C₅ = ________,
C₆ = ________

Question 30

Consider this initial-value problem: .
Assume a solution of this equation can be represented as a power series .
Write down the first four terms of the power series solution.

Question 31

Consider the second-order differential equation .
Assume the solution can be expressed as a power series . Assume C₀ = 0. Find C₁

Question 32

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

Question 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) +7(X +16) - 8xy = 0 about the point ?

Question 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) + 8(x + 7) - 7xy = 0 about the point ?

Question 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) + 6(x + 14) - 2xy = 0 about the point X₀ = 11?

Question 36

What is the radius of convergence of a series solution for the second-order differential equation .

A) 0
B) 1
C) .
D) $\infty$

Question 37

What is a lower bound for the radius of convergence of a series solution for the second-order differential equation .

Question 38

What is the greatest lower bound of the radius of convergence of a series solution for the second-order differential equation .

A) $\sqrt{15}$
B) $\frac{1}{4}$
C) $\frac{\sqrt{15}}{4}$
D) $\frac{1}{2}$

Question 39

Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation .

A) $\left\{x^{-3}, x^{3}\right\}$
B) $\left\{x^{3}, x^{6}\right\}$
C) $\left\{x^{3}, x^{3} \ln x\right\}$
D) $\left\{x^{-3}, x^{-3} \ln x\right\}$
E) $\left\{x^{3} \cos (\ln x), x^{3} \sin (\ln x)\right\}$

Question 40

Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation .

A) $\left\{x^{-5}, x^{-5} \cos (\ln x)\right\}$
B) $\left\{x^{5}, x^{-5}\right\}$
C) $\left\{x^{5}, x^{5} \ln x\right\}$
D) $\left\{x^{-5}, x^{-5} \ln x\right\}$
E) $\left\{x^{5}, x^{5} \cos (\ln x)\right\}$

Question 41

Which of the following pairs forms a fundamental set of solutions of the Cauchy Euler differential equation .

A) $\{\cos (6 \ln x), \sin (6 \ln x)\}$
B) $\left\{x^{6} \cos (\ln x), x^{6} \sin (\ln x)\right\}$
C) $\{\ln (\cos (6 x)), \ln (\sin (6 x))\}$
D) $\left\{x^{-6}, x^{6}\right\}$
E) $\{\cos (\ln (6 x)), \sin (\ln (6 x))\}$

Question 42

Find the general solution of the Cauchy Euler differential equation .

A) $y=C_{1} x^{-\frac{1}{2}}+C_{2} x^{\frac{1}{2}}$
B) $y=C_{1} x^{\frac{1}{2}}+C_{2} x^{\frac{1}{2}} \ln x$
C) $y=C_{1} x^{-\frac{1}{2}}+C_{2} x^{-\frac{1}{2}} \ln x$
D) $y=x^{-\frac{1}{2}}\left(C_{1} \sin \frac{1}{2} \ln x+C_{2} \cos \frac{1}{2} \ln x\right)$
E) $y=x^{-\frac{1}{2}}\left(C_{1} \sin \ln \frac{1}{2} x+C_{2} \cos \ln \frac{1}{2} x\right)$
F) $y=x^{\frac{1}{2}}\left(C_{1} \sin -\frac{1}{2} \ln x+C_{2} \cos -\frac{1}{2} \ln x\right)$

Question 43

Find the general solution of the Cauchy Euler differential equation .

A) $y=C_{1} x^{-\frac{8}{5}}+C_{2} x^{-\frac{8}{3}}$
B) $y=C_{1} x^{\frac{8}{5}}+C_{2} x^{\frac{8}{3}}$
C) $y=C_{1} x^{\frac{5}{8}}+C_{2} x^{\frac{5}{8}} \ln x$
D) $y=C_{1} x^{\frac{8}{3}}+C_{2} x^{\frac{8}{3}} \ln x$
E) $y=C_{1} x^{\frac{8}{5}}+C_{2} x^{-\frac{8}{3}}$

Question 44

Find the general solution of the Cauchy Euler differential equation .

A) $y=C_{1} x^{-2}+C_{2} x^{-7}$
B) $y=C_{1} x^{2}+C_{2} x^{-7}$
C) $y=C_{1} x^{-2}+C_{2} x^{7}$
D) $y=C_{1} x^{2}+C_{2} x^{7}$

Question 45

Solve this initial value problem: .

A) $y=\frac{25}{9} x^{6}+\frac{25}{9} x^{-3}$
B) $y=\frac{25}{9} x^{-6}+\frac{25}{9} x^{3}$
C) $y=-\frac{25}{3} x^{6}-\frac{40}{3} x^{-3}$
D) $y=-\frac{25}{3} x^{-6}-\frac{40}{3} x^{3}$

Question 46

Solve this initial value problem: .
.

Question 47

Find the general solution of the Cauchy Euler differential equation .

A) $y=C_{1}(x-3)^{5}+C_{2}(x-3)^{5} \ln x$
B) $y=C_{1}(x-3)^{-5}+C_{2}(x-3)^{5}$
C) $y=C_{1}(x-3)^{-5}+C_{2}(x-3)^{-5} \ln x$
D) $y=C_{1}(x-3)^{5}+C_{2}(x-3)^{10}$

Question 48

Find the general solution of the Cauchy Euler differential equation .

A) $y=(x+1)^{-7}\left(C_{1}+C_{2} \ln x\right)$
B) $y=C_{1}(x+1)^{-7}+C_{2}(x+1)^{7}$
C) $y=(x+1)^{-7}\left(C_{1} \sin (\ln x)+C_{2} \cos (\ln x)\right)$
D) $y=(x+1)^{7}\left(C_{1}+C_{2} \ln x\right)$
E) $y=(x+1)^{7}\left(C_{1} \ln (\sin x)+C_{2} \ln (\cos x)\right)$

Question 49

Consider the Bessel equation of order .
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.

Question 50

Consider the Legendre equation: .
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.

Question 51

Consider the second-order differential equation .
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.

Question 52

Consider the second-order differential equation: .
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.

Question 53

Consider the second-order differential equation .
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.

Question 54

x = 0 is a regular singular point for the second-order differential equation
.

Question 55

Consider the second-order differential equation .
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.

Question 56

Consider the second-order differential equation: .
Why is C₀ = 0 a regular singular point?

A) The functions $x^{2} \cdot \frac{7 x(x+1)}{5 x^{2}}$ and $x \cdot\left(-\frac{7}{5 x^{2}}\right)$ both have convergent Taylor series expansions about 0 .
B) The functions $x \cdot \frac{7 x(x+1)}{5 x^{2}}$ and $x^{2} \cdot\left(-\frac{7}{5 x^{2}}\right)$ both have convergent Taylor series expansions about 0 .
C) $\lim _{x \rightarrow 0} x \cdot\left(-\frac{7}{5 x^{2}}\right)=\infty$
D) $\lim _{x \rightarrow 0} x \cdot\left(-\frac{7}{5 x^{2}}\right) \neq 0$

Question 57

Consider the second-order differential equation: .
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

Question 58

Consider the second-order differential equation: .
Which of these is the recurrence relation for the coefficients?

A) $a_{n}=\frac{-2(n+r-1) a_{n-1}}{(2(n+r)+5)(n+r+1)}, n \geq 1$
B) $a_{n}=\frac{2(n+r-1) a_{n-1}}{(2(n+r)-5)(n+r+1)}, n \geq 1$
C) $a_{n}=\frac{-2(n+r-1) a_{n-1}}{(2(n+r)-1)(n+r+5)}, n \geq 1$
D) $a_{n}=\frac{2(n+r-1) a_{n-1}}{(2(n+r)+1)(n+r-5)}, n \geq 1$

Question 59

Consider the second-order differential equation: .
Write out the first three terms of the solution corresponding to the positive root of the indicial equation.
Y₁ (x) ≈ ________

Question 60

Consider the second-order differential equation: .
Write out the first three terms of the solution corresponding to the nonpositive root of the indicial equation.
Y₂ (x) ≈ ________

Question 61

Consider the second-order differential equation: .
The general solution of the differential equation is .
are arbitrary real constants.

Question 62

Consider the Bessel equation of order .
Suppose the method of Frobenius is used to determine a power series solution of the form .
Of this differential equation. Assume a₀ $\neq$ 0.
Which of these is the indicial equation?

A) r² - 36 = 0
B) r²- 6 = 0
C) r²+ 6 = 0
D) r²+ 36 = 0

Question 63

Consider the Bessel equation of order .
Suppose the method of Frobenius is used to determine a power series solution of the form .
Of this differential equation. Assume a₀ $\neq$ 0.
Which of these is the recurrence relation for the coefficients?

A) $a_{1}=0, a_{n+2}=\frac{-a_{n}}{(r+n-2)^{2}+16}, n \geq 0$
B) $a_{1}=0, a_{n+2}=\frac{-a_{n}}{(r+n+2)^{2}-16}, n \geq 0$
C) $a_{1}=1, a_{n+2}=\frac{-a_{n}}{(r+n-2)^{2}+16}, n \geq 0$
D) $a_{1}=1, a_{n+2}=\frac{-a_{n}}{(r+n+2)^{2}-16}, n \geq 0$

Question 64

Consider the Bessel equation of order .
Suppose the method of Frobenius is used to determine a power series solution of the form .
Of this differential equation. Assume a₀ $\neq$ 0.
Which of these is the explicit formula for the coefficients corresponding to the positive root of the indicial equation?

A) $a_{2 n}=0$ and $a_{2 n+1}=(-1)^{n} \cdot \frac{a_{0} \cdot 4 !}{2^{n+1}(n+1) !(n+4) !}, n \geq 1$
B) $a_{2 n}=0$ and $a_{2 n+1}=(-1)^{n} \cdot \frac{a_{0} \cdot 4 !}{2^{2 n} n !(n+4) !}, n \geq 1$
C) $a_{2 n+1}=0$ and $a_{2 n}=(-1)^{n-1} \cdot \frac{a_{0} \cdot 4 !}{2^{n} n !(n+4) !}, n \geq 1$
D) $a_{2 n+1}=0$ and $a_{2 n}=(-1)^{n} \cdot \frac{a_{0} \cdot 4 !}{2^{2 n} n !(n+4) !}, n \geq 1$

Question 65

Consider the Bessel equation of order .
Suppose the method of Frobenius is used to determine a power series solution of the form .
of this differential equation. Assume a₀ ≠ 0.
Write the power series solution corresponding to the positive root of the indicial equation.
Y₁ (x) = ________

Question 66

Consider the second-order differential equation .
Write the differential equation in the form . a regular singular point for this equation?

A) $x^{2} p(x)$ and $x q(x)$ both have convergent Taylor expansions about 0 .
B) $\lim _{x \rightarrow 0} x^{2} p(x)=0$ and $\lim _{x \rightarrow 0} x q(x)=\infty$
C) $\lim _{x \rightarrow 0} x p(x)$ is finite and $\lim _{x \rightarrow 0} x^{2} \gamma(x)=\infty$
D) $\lim _{x \rightarrow 0} x p(x)$ is finite and $x^{2} \gamma(x)$ has a convergent Taylor expansion about 0 .

Question 67

Consider the second-order differential equation .
Suppose the method of Frobineius is used to determine a power series solution of the form .
Of this differential equation. Assume a₀ $\neq$ 0. Which of these is the indicial equation?

A) r²+ 7r = 0
B) + 8r = 0
C) - 8r = 0
D) - 7r = 0

Question 68

Consider the second-order differential equation .
Suppose the method of Frobenius is used to determine a power series solution of the form .
Of this differential equation. Assume a₀ $\neq$ 0.
Which of these is the indicial equation?

A) r² + r = 0
B) r² - r = 0
C) r² + r - 2 = 0
D) r² - r - 2 = 0

Question 69

Consider the second-order differential equation .
Suppose the method of Frobenius is used to determine a power series solution of the form .
of this differential equation. Assume a₀ ≠ 0.
Using the larger root of the indicial equation, write down an explicit formula for the coefficients and the corresponding power series solution.
a_n = ________, n ≥ 1
y₁ (x) = ________

Question 70

Consider the second-order differential equation .
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 .
Assume a₀ $\neq$ 0.
Which of these is the recurrence relation for the coefficients?

A) $a_{1}=0, a_{n}=-\frac{64 \cdot a_{n-2}}{2^{n}}, n \geq 2$
B) $a_{1}=1, a_{n}=\frac{64 \cdot a_{n-2}}{2^{n}}, n \geq 2$
C) $a_{1}=1, a_{n}=\frac{64 \cdot a_{n-2}}{n^{2}}, n \geq 2$
D) $a_{1}=0, a_{n}=-\frac{64 \cdot a_{n-2}}{n^{2}}, n \geq 2$
E) $a_{1}=0, a_{n}=-\frac{64 \cdot a_{n-2}}{n}, n \geq 2$

Question 71

Consider the second-order differential equation .
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 .
Assume a₀ $\neq$ 0.
Which of these is the explicit formula for the coefficients ?

A) $a_{2 n}=\frac{(-1)^{n} 2^{3 n}}{n !} a_{0, n} \geq 1$
B) $a_{2 n}=\frac{(-1)^{n} 2^{5 n}}{n !} a_{0}, n \geq 1$
C) $a_{2 n}=\frac{(-1)^{n} 2^{3 n}}{(n !)^{2}} a_{0}, n \geq 1$
D) $a_{2 n}=\frac{(-1)^{n} 2^{5 n}}{(n !)^{2}} a_{0}, n \geq 1$

Question 72

Consider the second-order differential equation .
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 .
Assume a₀ $\neq$ 0.
Assuming that a₀ = 1, one solution of the given differential equation is .
Assume x > 0. Which of these is a form of a second solution of the given differential equation, linearly independent to (x)?

A) $y_{2}(x)=y_{1}(x) \ln x+\sum_{n=1}^{\infty} a^{*}{ }_{n} x^{n}$
B) $y_{2}(x)=y_{1}^{\prime}(x) \ln x \mid+\sum_{n=1}^{\infty} a^{*} n^{n}$
C) $y_{2}(x)=\ln x+y_{1}(x) \cdot \sum_{n=1}^{\infty} a^{*}{ }_{n} x^{n}$
D) $y_{2}(x)=|\ln x|+y_{1}(x) \cdot \sum_{n=1}^{\infty} a^{n}{ }_{n} x^{n}$
E) $y_{2}(x)=\ln x\left(y_{1}(x)+\sum_{n=1}^{\infty} a^{n}{ }_{n} x^{n}\right)$

Question 73

Consider the second-order differential equation .
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 Assume a₀ $\neq$ 0.
Assuming that a₀ = 1, one solution of the given differential equation is
Differentiating as needed, which of these relationships is correct?

A) $2 y^{\prime}(x)-a_{1}^{*}-128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}-128 a_{n-2}^{*}\right) x^{n-1}=0$
B) $2 y^{\prime}(x)+a_{1}^{*}+128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}+128 a_{n-2}^{*}\right) x^{n-1}=0$
C) $2 y^{\prime}(x)+a_{1}^{*}-128 a_{2}^{*} x-\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}+128 a_{n-2}^{*}\right) x^{n-1}=0$
D) $2 y^{\prime}(x)+a_{1}^{*}+128 a_{2}^{*} x+\sum_{n=3}^{\infty}\left(n^{2} a_{n}^{*}-128 a_{n-2}^{*}\right) x^{n-1}=0$

Question 74

Consider the second-order differential equation .
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 . Assume a₀ ≠ 0.
Assuming that a₀= 1, one solution of the given differential equation is
Assuming that are known, what is the radius of convergence of the power series of the second solution Y₂ (x)?

Question 75

Consider the second-order differential equation .
Using the method of Frobenius, which of these is the general solution of this differential equation? Assume are arbitrary real constants.

A) $y(x)=a_{0}+a_{0}^{*}\left(x \ln x+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n+1}}{n \cdot n !}\right)$
B) $y(x)=a_{0} x+a_{0}^{*}\left(x \ln x+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n+1}}{n \cdot n !}\right)$
C) $y(x)=a_{0}+a^{*}{ }_{0}^{x}\left[\ln x+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n+1}}{n \cdot n !}\right)$
D) $y(x)=a_{0}+a_{0}^{*} \ln x\left(1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{n+1}}{n \cdot n !}\right)$

Question 76

Consider the second-order differential equation .
What is the radius of convergence of the series of the general solution of the differential equation?

A) 1
B) 2
C) 4
D) ?

Question 77

Consider the second-order differential equation .
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+ = 0
B) + r - = 0
C) - r + = 0
D) - r - = 0

Question 78

Consider the second-order differential equation .
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

A) $y_{1}(x)=\sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=x^{-\frac{1}{4}} \sum_{n=0}^{\infty} b_{n} x^{n}$
B) $y_{1}(x)=\sum_{n=0}^{\infty} a_{n} x^{n+1}, y_{2}(x)=x^{-\frac{1}{4}} \sum_{n=0}^{\infty} b_{n} x^{n}$
C) $y_{1}(x)=\sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=x^{\frac{3}{4}} \sum_{n=0}^{\infty} b_{n} x^{n}$
D) $y_{1}(x)=\sum_{n=0}^{\infty} a_{n} x^{n+1}, y_{2}(x)=x^{\frac{3}{4}} \sum_{n=0}^{\infty} b_{n} x^{n}$
E) $y_{1}(x)=\ln (x) \sum_{n=0}^{\infty} a_{n} x^{n+1}, y_{2}(x)=x^{-\frac{1}{4}} \sum_{n=0}^{\infty} b_{n} x^{n}$

Question 79

Consider the second-order differential equation .
Which of these is the indicial equation about the regular singular point x = 0?

A) r² - 9r - 16 = 0
B) r² - 7r + 16 = 0
C) r² + 7r + 16 = 0
D) + 8r - 16 = 0
E) - 8r + 16 = 0

Question 80

Consider the second-order differential equation .
Which of the following is the form of a pair of linearly independent solutions of this differential equation?

A) $y_{1}(x)=x^{-4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=\ln x\left(y_{1}(x)+x^{-4} \sum_{n=0}^{\infty} b_{n} x^{n}\right)$
B) $y_{1}(x)=x^{4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=\ln x\left(y_{1}(x)+x^{4} \sum_{n=0}^{\infty} b_{n} x^{n}\right)$
C) $y_{1}(x)=x^{-4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=y_{1}(x) \ln x+x^{-4} \sum_{n=0}^{\infty} b_{n} x^{n}$
D) $\left.y_{1}(x)=x^{4} \sum_{n=0}^{\infty} a_{n} x^{n}, y_{2}(x)=y_{1} x\right) \ln x+x^{4} \sum_{n=0}^{\infty} b_{n} x^{n}$