Question 1

Find the general solution of the Cauchy Euler differential equation   .

Accepted Answer

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}} $ 
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 2

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?

Accepted Answer

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

Question 3

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)?

Accepted Answer

Question 4

Find the general solution of the Cauchy Euler differential equation   .

Accepted Answer

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} $ 
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 5

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

Accepted Answer

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

Question 6

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 = ________

Accepted Answer

Question 7

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?

Accepted Answer

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 $ 
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 8

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?

Accepted Answer

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 $ 
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 9

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

Accepted Answer

A)  (3r - 1)(r + 5) = 0 
B)  (3r + 1)(r - 5) = 0 
C)  (3r + 5)(r - 1) = 0 
D)  (3r - 5)(r + 1) = 0 
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 10

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

Accepted Answer

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

Question 11

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 ?

Accepted Answer

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 $ 
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 12

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   ?

Accepted Answer

The answer of What is a lower bound for the...

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

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

Accepted Answer

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. 
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 15

Find the general solution of the Cauchy Euler differential equation   .

Accepted Answer

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) $ 
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 16

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

Accepted Answer

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 $ 
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 17

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

Accepted Answer

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

Question 18

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) = ________

Accepted Answer

Question 19

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 ?

Accepted Answer

The answer of What is the greatest lower bound for...

Question 20

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 Cn
   = , n = 0, 1, 2, ...

Accepted Answer

Find the general solution of the Cauchy Euler differential equation .

Find the general solution of the Cauchy Euler differential equation .

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

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 =

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

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

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 ?

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

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

Find the general solution of the Cauchy Euler differential equation .

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

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

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) = ________

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 ?

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, ...

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

Find the general solution of the Cauchy Euler differential equation .

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?

Find the general solution of the Cauchy Euler differential equation .

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

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 = ________

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?

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

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

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 ?

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

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

Find the general solution of the Cauchy Euler differential equation .

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

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

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) = ________

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 ?

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 Cn = , n = 0, 1, 2, ...

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

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 =

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) = ________

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, ...