Question 1

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

Accepted Answer

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))\} $ 
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 2

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) ≈ ________

Accepted Answer

Question 3

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?

Accepted Answer

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} $ 
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 4

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?

Accepted Answer

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 $ 
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 5

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

Accepted Answer

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\} $ 
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 6

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 Cn? Assume that C0 is known

Accepted Answer

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 $ 
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 7

Find the general solution of the Cauchy Euler differential equation   .

Accepted Answer

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) $ 
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 8

Consider the Legendre equation:   .
Which of these statements is true?

Accepted Answer

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. 
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 9

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

Accepted Answer

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. 
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 10

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

Accepted Answer

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 . 
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 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.
Assuming that a0 = 1, one solution of the given differential equation is  
Differentiating as needed, which of these relationships is correct?

Accepted Answer

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 $ 
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 12

What is the radius of convergence of the power series   ?

Accepted Answer

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

Question 13

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

Accepted Answer

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 $ 
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 14

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 Cn ? Assume that C0 is known.

Accepted Answer

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 $ 
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 15

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) ≈ ________

Accepted Answer

Question 16

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

Accepted Answer

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

Question 17

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) = ________

Accepted Answer

@#IMG-DLM& (x) = sin

Question 18

Solve this initial value problem:   .

Accepted Answer

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} $ 
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 19

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

Accepted Answer

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. 
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 20

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.

Accepted Answer

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) $ 
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) $

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

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) ≈ ________

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?

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

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

Find the general solution of the Cauchy Euler differential equation .

Consider the Legendre equation: . Which of these statements is true?

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

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

What is the radius of convergence of the power series ?

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

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.

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) ≈ ________

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

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) =

Solve this initial value problem: .

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

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.

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

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

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) ≈ ________

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?

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

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 Cn? Assume that C0 is known

Find the general solution of the Cauchy Euler differential equation .

Consider the Legendre equation: . Which of these statements is true?

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

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

What is the radius of convergence of the power series ?

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

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 Cn ? Assume that C0 is known.

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) ≈ ________

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

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) = ________

Solve this initial value problem: .

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

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.

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 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) ≈ ________

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

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

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.

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) ≈ ________

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) =