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

Accepted Answer

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 Cn 
   .

Accepted Answer

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

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:   ....

Question 7

What is the radius of convergence of the power series   ?

Accepted Answer

A)  5 
B)  6 
C)  36 
D)  $\infty$ 
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)  7
F) 8 
A)  -5 
B)  -7 
C)  5 
D)  6 
E)  7
F) 8

Question 9

Find the general solution of the Cauchy Euler differential equation   .

Accepted Answer

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} $ 
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 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. 
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^{\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}} $ 
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 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)  $\infty$ 
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 
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

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.
an = ________, n ≥ 1
y1 (x) = ________

Accepted Answer

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

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 ?

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?

Which of these power series is equivalent to + 7 ?

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

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 ?

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

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