Question 1

Consider the initial value problem

Which of these statements are true? Select all that apply.
A) There exists a nonzero real number r such that y(t) =  is a solution of the initial value problem.
B) This initial value problem has only one solution on the interval (-7, 5).
C) The constant function y(t) = -1 is a solution of this initial value problem for all real numbers t.
D) There must exist a function y = ?(t) that satisfies this initial value problem on the interval  .
E) The constant function y(t) = 0 is the unique solution of this initial value problem on the interval  .

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Which of these...

Question 2

Suppose that Y₁ and Y₂ are both solutions of the differential equation . Which of the following must also be solutions of this differential equation? Select all that apply. Here, C₁ , and C₂ are arbitrary real constants. A) $ 5 y_{1}-4 y_{2} $ B) $ t y_{1} $ C) $ C_{1} $ D) $ \left(C_{1} y_{1}\right) \cdot\left(C_{2} y_{2}\right) $ E) $ C_{1}\left(y_{1}+y_{2}\right) $ F) $ C_{1}\left(7 y_{1}-9 y_{2}\right)-C_{2}\left(2 y_{1}-7 y_{2}\right. $

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Question 3

If Y₁ and Y₂ are both solutions of the differential equation then Y₁ - Y₂ is also a solution of this equation.

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Question 4

Consider the differential equation Which of the following statements is true? A) If 2 is a solution of this differential equation, then so is . B) If Y₁ and Y₂ are both solutions of this differential equation, then Y₁ - Y₂ cannot be a solution of it. C) The Principle of Superposition guarantees that if y₁ and y₂ are both solutions of this differential equation, then C₁ y₁ + C₂ y₂ must also be a solution of it, for any choice of real constants and . D) There exist nonzero real constants C₁ and C₂ such that C₁ y₁ - C₂ y₂ is a solution of this differential equation.

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Which of...

Question 5

Compute the Wronskian of the pair of functions 
A) -2 
B) -6 
C) -8 
D) -6
E) -8

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Question 6

Compute the Wronskian of the pair of functions sin(5t) and cos(5t).
A) -5
B) -4
C) 1
D) 4
E) 5

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Question 7

Compute the Wronskian of the pair of functions 2t  and 4  .

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Question 8

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Compute the Wronskian of this function pair. A) $ \frac{1}{t^{2}} $ B) $ \frac{1}{1} $ C) $ \frac{\ln t}{t} $ D) $ (\ln t)^{2} $ E) $ \ln \left(t^{2}\right) $

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Question 9

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Which of these statements is true? A) Both y₁ and y₂ can be solutions of the differential equation on the interval (0, $\infty$), where p(t) and q(t) are continuous on (0, $\infty$). B) The Wronskian for this function pair is strictly positive on (0, $\infty$). C) Abel's theorem implies that y₁ and y₂ cannot both be solutions of any differential equation of the form on the interval (0, $\infty$). D) The pair y₁ and y₂ constitutes a fundamental set of solutions to some second-order differential equation of the form on the interval (0, $\infty$).

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Question 10

Consider the pair of functions y₁ = t and y₂ = 3t². Which of these statements are true? Select all that apply. A) W[y₁ , y₂](t) > 0 for all values of t in the interval (-2, 2). B) W[y₁ , y₁](t) = 3t² C) The pair y₁ and y₂ constitutes a fundamental set of solutions to some second-order differential equation of the form on the interval (-2, 2). D) Abel's theorem implies that y₁ and y₂ cannot both be solutions of any differential equation of the form on the interval (-2, 2). E) Since there exists a value of t₀ in the interval (-2, 2) for which W[y₁ ,y₂ ](t) = 0, there must exist a differential equation of the form for which the pair y₁ and y₂ constitute a fundamental set of solutions on the interval (-2, 2).

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Question 11

Which of these is a fundamental set of solutions for the differential equation  
Select all that apply.
A) $ y_{1}=\cos (10 t) $ and $ y_{2}=\sin (10 t) $
B) $ y_{3}=7 \cos (10 t)-20 \sin (10 t) $ and $ y_{4}=10 \cos (10 t)-14 \sin (10 t) $
C) $ y_{5}=e^{-10 t} $ and $ y_{6}=e^{10 t} $
D) $ y_{7}=e^{10 t} \sin (10 t) $ and $ y_{8}=e^{10 t} \cos (10 t) $
E) $ y_{9}=7 \sin (10 t) $ and $ y_{10}=7 \sin (10 t)-8 \cos (10 t) $

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Question 12

The pair of functions  
forms a fundamental set of solutions for the differential equation

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forms a...

Question 13

What is the characteristic equation for the second-order homogeneous differential equation  + 36y=0?
A) $ (r-6)(r+6)=0 $
B) $ r^{2}+6=0 $
C) $ r^{2}+36=0 $
D) $ r^{2}+36 r=0 $

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Question 14

For which of these differential equations is the characteristic equation given by  
A) $ y^{\prime \prime}+50=0 $
B) $ y^{\prime \prime}+50 y=0 $
C) $ y^{\prime \prime}-2 y^{\prime}+50=0 $
D) $ y^{\prime \prime}-2 y^{\prime}+50 y=0 $
E) $ \left(y^{\prime}-(1-7 i)\right)\left(y^{\prime}-(1+7 i)\right)=0 $
F) $ \left(y^{\prime}-(1-7 i) y\right)\left(y^{\prime}-(1+7 i) y\right)=0 $

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Question 15

Which of the following are solutions to the homogeneous second-order differential equation 
 Select all that apply.
A) $ y_{1}=2 \sin \left(\frac{6}{7} t\right) $
B) $ y_{2}=C\left(\cos \frac{6}{7} t+\sin \frac{6}{7} t\right) $, where $ C $ is any real constant
C) $ y_{3}=-2 \cos \left(\frac{7}{6} t\right) $
D) $ y_{4}=e^{\frac{6}{7} t} $
E) $ y_{5}=C_{1} e^{\frac{6}{7} t}+C_{2} e^{-\frac{6}{7} t} $ where $ C_{1} $ and $ C_{2} $ are any real constants
F) $ y_{6}=5 e^{\frac{7}{6} t}+7 e^{-\frac{7}{6} t} $
G) $ y_{7}=\sin \left(\frac{6}{7} t\right)+C $, where $ C $ is any real constant

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Question 16

Which of the following are solutions to the homogeneous second-order differential equation  
Select all that apply.
A) $ y_{1}=-\frac{1}{2} \pi \sin (3 t) $
B) $ y_{2}=e^{6 t} \cos (3 t) $
C) $ y_{3}=2 e^{6 t} $
D) $ y_{4}=5 e^{6 t}(\sin (3 t)+\cos (3 t)) $
E) $ y_{5}=C e^{-6 t} \cos (3 t) $, where $ C $ is any real constant
F) $ y_{6}=e^{-6 t} \cos (3 t) $

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Question 17

Which of the following is the general solution of the homogeneous second-order differential equation  
are arbitrary real constants.
A) $ y=C\left(\cos \frac{t}{3}+\sin \frac{t}{3}\right) $
B) $ y=C_{1} \cos (3 t)+C_{2} \sin (3 t) $
C) $ y=C(\cos (3 t)+\sin (3 t)) $
D) $ y=C_{1} \cos \left(\frac{t}{3}\right)+C_{2} \sin \left(\frac{t}{3}\right) $
E) $ y=\cos \left(\frac{t}{3}\right)+\sin \left(\frac{t}{3}\right)+C t $
F) $ y=\cos (3 t)+\sin (3 t)+C $

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Question 18

Which of the following is the general solution of the homogeneous second-order differential equation  
are arbitrary real constants.
A) $ y=C_{1} e^{4 t} \sin (6 t)+C_{2} e^{4 t} \cos (6 t) $
B) $ y=e^{-4 t}\left(C_{1} \sin (6 t)+C_{2} \cos (6 t)\right) $
C) $ y=C_{1} e^{4 t} \cos (6 t)+C_{2} e^{4 t} \sin (6 t)+C $
D) $ y=e^{6 t}(\sin (4 t)+\cos (6 t))+C $
E) $ y=C_{1} e^{-4 t} \sin (6 t)+C_{2} e^{-4 t} \cos (6 t)+C $

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Question 19

What is the solution of this initial value problem:  
A) $ y=3 \cos (11 t)+\frac{10}{11} \sin (11 t) $
B) $ y=\cos (11 t)+\sin (11 t) $
C) $ y=3 \sin (11 t)+10 \cos (11 t) $
D) $ y=10 e^{-11 t}+3 e^{-11 t} $
E) $ y=3 e^{121 t}+\frac{10}{11} e^{-121 t} $

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Question 20

What is the solution of this initial value problem:

A) $ y=e^{3 t}\left[3 \cos (7 t)-\frac{4}{7} \sin (7 t)\right] $
B) $ y=e^{-3 t}(3 \cos (7 t)+2 \sin (7 t)) $
C) $ y=e^{7 t}\left(3 \cos (3 t)+\frac{14}{3} \sin (3 t)\right) $
D) $ y=e^{-7 t}\left[3 \cos (3 t)-\frac{4}{3} \sin (3 t)\right] $

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Quiz 3: Second-Order Linear Differential Equations

Consider the initial value problem Which of these statements are true? Select all that apply.

If Y₁ and Y₂ are both solutions of the differential equation then Y₁ - Y₂ is also a solution of this equation.

Consider the differential equation Which of the following statements is true?

Compute the Wronskian of the pair of functions

Compute the Wronskian of the pair of functions sin(5t) and cos(5t) .

Compute the Wronskian of the pair of functions 2t and 4 .

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Compute the Wronskian of this function pair.

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Which of these statements is true?

Consider the pair of functions y₁ = t and y₂ = 3t². Which of these statements are true? Select all that apply.

The pair of functions forms a fundamental set of solutions for the differential equation

What is the characteristic equation for the second-order homogeneous differential equation $What is the characteristic equation for the second-order homogeneous differential equation + 36y=0? A) (r-6) (r+6) =0 B) r^{2}+6=0 C) r^{2}+36=0 D) r^{2}+36 r=0$ + 36y=0?

What is the solution of this initial value problem: $What is the solution of this initial value problem: A) y=3 \cos (11 t) +\frac{10}{11} \sin (11 t) B) y=\cos (11 t) +\sin (11 t) C) y=3 \sin (11 t) +10 \cos (11 t) D) y=10 e^{-11 t}+3 e^{-11 t} E) y=3 e^{121 t}+\frac{10}{11} e^{-121 t}$

Quiz 3: Second-Order Linear Differential Equations

Consider the initial value problem Which of these statements are true? Select all that apply.

Suppose that Y1 and Y2 are both solutions of the differential equation . Which of the following must also be solutions of this differential equation? Select all that apply. Here, C1 , and C2 are arbitrary real constants.

If Y1 and Y2 are both solutions of the differential equation then Y1 - Y2 is also a solution of this equation.

Consider the differential equation Which of the following statements is true?

Compute the Wronskian of the pair of functions

Compute the Wronskian of the pair of functions sin(5t) and cos(5t) .

Compute the Wronskian of the pair of functions 2t and 4 .

Consider the pair of functions y1 = ln t and y1 = t ln t. Compute the Wronskian of this function pair.

Consider the pair of functions y1 = ln t and y1 = t ln t. Which of these statements is true?

Consider the pair of functions y1 = t and y2 = 3t2. Which of these statements are true? Select all that apply.

Which of these is a fundamental set of solutions for the differential equation Select all that apply.

The pair of functions forms a fundamental set of solutions for the differential equation

What is the characteristic equation for the second-order homogeneous differential equation + 36y=0?

For which of these differential equations is the characteristic equation given by

Which of the following are solutions to the homogeneous second-order differential equation Select all that apply.

Which of the following are solutions to the homogeneous second-order differential equation Select all that apply.

Which of the following is the general solution of the homogeneous second-order differential equation are arbitrary real constants.

Which of the following is the general solution of the homogeneous second-order differential equation are arbitrary real constants.

What is the solution of this initial value problem:

What is the solution of this initial value problem:

If Y₁ and Y₂ are both solutions of the differential equation then Y₁ - Y₂ is also a solution of this equation.

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Compute the Wronskian of this function pair.

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Which of these statements is true?

Consider the pair of functions y₁ = t and y₂ = 3t². Which of these statements are true? Select all that apply.

What is the characteristic equation for the second-order homogeneous differential equation $What is the characteristic equation for the second-order homogeneous differential equation + 36y=0? A) (r-6) (r+6) =0 B) r^{2}+6=0 C) r^{2}+36=0 D) r^{2}+36 r=0$ + 36y=0?

What is the solution of this initial value problem: $What is the solution of this initial value problem: A) y=3 \cos (11 t) +\frac{10}{11} \sin (11 t) B) y=\cos (11 t) +\sin (11 t) C) y=3 \sin (11 t) +10 \cos (11 t) D) y=10 e^{-11 t}+3 e^{-11 t} E) y=3 e^{121 t}+\frac{10}{11} e^{-121 t}$