Question 1

Solve this initial value problem:  . 
  .

Accepted Answer

y = 2 cos(8π ln x) -  sin(8π ln x)

Question 2

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?

Accepted Answer

5

Question 3

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, ...

Accepted Answer

The answer of Consider the second-order differential equation  
Assume...

Question 4

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 .

Accepted Answer

The answer of Consider the second-order differential equation  -...

Question 5

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).

Accepted Answer

The answer of Consider the first-order differential equation  
Assume...

Question 6

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.

Accepted Answer

The answer of Consider this initial-value problem:  . 
Assume...

Question 7

What is the radius of convergence of the power series  ?

Accepted Answer

A)   5
B)   6
C)   36
D)    ∞\infty∞

Question 8

Which of these are singular points for the differential equation

Select all that apply.

Accepted Answer

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

Question 9

Find the general solution of the Cauchy Euler differential equation  .

Accepted Answer

A)    y=C1(x−3) 5+C2(x−3) 5ln⁡x y=C_{1}(x-3) ^{5}+C_{2}(x-3) ^{5} \ln x y=C1​(x−3) 5+C2​(x−3) 5lnx 
B)    y=C1(x−3) −5+C2(x−3) 5 y=C_{1}(x-3) ^{-5}+C_{2}(x-3) ^{5} y=C1​(x−3) −5+C2​(x−3) 5 
C)   y=C1(x−3) −5+C2(x−3) −5ln⁡x y=C_{1}(x-3) ^{-5}+C_{2}(x-3) ^{-5} \ln x y=C1​(x−3) −5+C2​(x−3) −5lnx 
D)    y=C1(x−3) 5+C2(x−3) 10 y=C_{1}(x-3) ^{5}+C_{2}(x-3) ^{10} y=C1​(x−3) 5+C2​(x−3) 10

Question 10

Consider the second-order differential equation:  .
Which of these statements is true?

Accepted Answer

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 11

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.

Accepted Answer

A)    y(x) =e(9x) 22 y(x) =e^{\frac{(9 x) ^{2}}{2}} y(x) =e2(9x) 2​ 
B)    y(x) =e92x2 y(x) =e^{\frac{9}{2} x^{2}} y(x) =e29​x2 
C)    y(x) =9ex22 y(x) =9 e^{\frac{x^{2}}{2}} y(x) =9e2x2​ 
D)    y(x) =92ex2 y(x) =\frac{9}{2} e^{x^{2}} y(x) =29​ex2

Question 12

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

Accepted Answer

A)   0
B)   1
C)    .
D)    ∞\infty∞

Question 13

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?

Accepted Answer

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

Question 14

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

Accepted Answer

The answer of Consider the second-order differential equation  ....

Question 15

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) = ________

Accepted Answer

The answer of Consider the second-order differential equation  ....

Question 16

Which of these are singular points for the differential equation

Select all that apply.

Accepted Answer

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

Question 17

What is the radius of convergence of the power series  ?

Accepted Answer

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

Question 18

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

Accepted Answer

A)   
B)   
C)   
D)

Question 19

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?

Accepted Answer

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

Question 20

Which of these power series is equivalent to  + 7  ?

Accepted Answer

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

Exam 5: Series Solutions of Second-Order Linear Equations

Solve this initial value problem: . .

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?

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, ...

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 .

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).

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.

What is the radius of convergence of the power series $What is the radius of convergence of the power series ? A) 5 B) 6 C) 36 D) \infty$ ?

Which of these are singular points for the differential equation Select all that apply.

Consider the second-order differential equation: . Which of these statements is true?

What is the radius of convergence of a series solution for the second-order differential equation $What is the radius of convergence of a series solution for the second-order differential equation . A) 0 B) 1 C) . D) \infty$ .

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?

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

Which of these are singular points for the differential equation Select all that apply.

What is the radius of convergence of the power series ?

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

Which of these power series is equivalent to + 7 ?

Exam 5: Series Solutions of Second-Order Linear Equations

Solve this initial value problem: . .

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?

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 Cn: C2n= ________, n = 0, 1, 2, ... C2n+1= ________, n = 0, 1, 2, ...

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 Cn .

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).

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.

What is the radius of convergence of the power series ?

Which of these are singular points for the differential equation Select all that apply.

Find the general solution of the Cauchy Euler differential equation .

Consider the second-order differential equation: . Which of these statements is true?

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.

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

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?

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

Which of these are singular points for the differential equation Select all that apply.

What is the radius of convergence of the power series ?

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

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?

Which of these power series is equivalent to + 7 ?

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?

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, ...

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 .

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).

What is the radius of convergence of the power series $What is the radius of convergence of the power series ? A) 5 B) 6 C) 36 D) \infty$ ?

What is the radius of convergence of a series solution for the second-order differential equation $What is the radius of convergence of a series solution for the second-order differential equation . A) 0 B) 1 C) . D) \infty$ .

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