Question 1

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

Accepted Answer

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 1 = 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) ." class="answers-bank-image d-inline" rel="preload" > 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 1 = 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) ." class="answers-bank-image d-inline" rel="preload" > 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 1 = 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) ." class="answers-bank-image d-inline" rel="preload" > for which the pair y1 and y2 constitute a fundamental set of solutions on the interval (-2, 2) . B, D

Question 2

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

Accepted Answer

A)   -5
B)   -4
C)   1
D)   4
E)   5 A

Question 3

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second.
What is the natural frequency? Round your answer to the nearest hundredth radian per second.

Accepted Answer

4.00 radians per second

Question 4

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

Accepted Answer

A)    y′′+50=0 y^{\prime \prime}+50=0 y′′+50=0 
B)    y′′+50y=0 y^{\prime \prime}+50 y=0 y′′+50y=0 
C)    y′′−2y′+50=0 y^{\prime \prime}-2 y^{\prime}+50=0 y′′−2y′+50=0 
D)    y′′−2y′+50y=0 y^{\prime \prime}-2 y^{\prime}+50 y=0 y′′−2y′+50y=0 
E)    (y′−(1−7i) ) (y′−(1+7i) ) =0 \left(y^{\prime}-(1-7 i) \right) \left(y^{\prime}-(1+7 i) \right) =0 (y′−(1−7i) ) (y′−(1+7i) ) =0 F)   (y′−(1−7i) y) (y′−(1+7i) y) =0 \left(y^{\prime}-(1-7 i)  y\right) \left(y^{\prime}-(1+7 i)  y\right) =0 (y′−(1−7i) y) (y′−(1+7i) y) =0

Question 5

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.

Accepted Answer

A)    5y1−4y2 5 y_{1}-4 y_{2} 5y1​−4y2​ 
B)    ty1 t y_{1} ty1​ 
C)    C1 C_{1} C1​ 
D)    (C1y1) ⋅(C2y2)  \left(C_{1} y_{1}\right)  \cdot\left(C_{2} y_{2}\right)  (C1​y1​) ⋅(C2​y2​)  
E)    C1(y1+y2)  C_{1}\left(y_{1}+y_{2}\right)  C1​(y1​+y2​)  F)   C1(7y1−9y2) −C2(2y1−7y2 C_{1}\left(7 y_{1}-9 y_{2}\right) -C_{2}\left(2 y_{1}-7 y_{2}\right. C1​(7y1​−9y2​) −C2​(2y1​−7y2​

Question 6

For which of these differential equations is the characteristic equation given by 6  + 7 = 0?

Accepted Answer

A)   6  + 7 = 0
B)   6  + 7 = 0
C)   6  + 7y = 0
D)   6  + 7y = 0

Question 7

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u₁ (t) and u₂ (t). u₁ (t) = ________ u₂(t) = ________

Accepted Answer

The answer of Consider the nonhomogeneous differential equation  Assume...

Question 8

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

Accepted Answer

A)    y1=8e−2t+2e2t y_{1}=8 e^{-2 t}+2 e^{2 t} y1​=8e−2t+2e2t 
B)    y2=Ce−2t y_{2}=\mathrm{Ce}^{-2 t} y2​=Ce−2t , where  C \mathrm{C} C  is any real constant
C)    y3=8(e2t+e−2t)  y_{3}=8\left(e^{2 t}+e^{-2 t}\right)  y3​=8(e2t+e−2t)  
D)    y4=Ce2t y_{4}=C e^{2 t} y4​=Ce2t , where  C C C  is any real constant
E)    y5=(C1e2t) ⋅(C2e−2t)  y_{5}=\left(C_{1} e^{2 t}\right)  \cdot\left(C_{2} e^{-2 t}\right)  y5​=(C1​e2t) ⋅(C2​e−2t)  , where  C1 C_{1} C1​  and  C2 C_{2} C2​  are any real constantsF)   y6=2e−2t y_{6}=2 e^{-2 t} y6​=2e−2t G)   y7=C(e−2t+e2t)  y_{7}=C\left(e^{-2 t}+e^{2 t}\right)  y7​=C(e−2t+e2t)  , where  C C C  is any real constant

Question 9

What is the general solution of the homogeneous second-order Cauchy Euler differential equation  are arbitrary real constants.

Accepted Answer

A)    y=C1t−6+C2t6 y=C_{1} t^{-6}+C_{2} t^{6} y=C1​t−6+C2​t6 
B)    y=C1(tln⁡t) −6+C2(tln⁡t) 6 y=C_{1}(t \ln t) ^{-6}+C_{2}(t \ln t) ^{6} y=C1​(tlnt) −6+C2​(tlnt) 6 
C)    y=t−6(C1+C2ln⁡t)  y=t^{-6}\left(C_{1}+C_{2} \ln t\right)  y=t−6(C1​+C2​lnt)  
D)    y=C1t−6+C2(tln⁡t) −6 y=C_{1} t^{-6}+C_{2}(t \ln t) ^{-6} y=C1​t−6+C2​(tlnt) −6

Question 10

Consider this second-order nonhomogeneous differential equation:

Which of these is a suitable form of a particular solution Y(t) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used? Here, all capital letters represent arbitrary real constants.

Accepted Answer

A)    Y(t) =Ae−3t Y(t) =A e^{-3 t} Y(t) =Ae−3t 
B)    Y(t) =eAt Y(t) =e^{A t} Y(t) =eAt 
C)    Y(t) =AeBt Y(t) =A e^{B t} Y(t) =AeBt 
D)    Y(t) =A+e−3t Y(t) =A+e^{-3 t} Y(t) =A+e−3t 
E)    Y(t) =Ae−3t+B Y(t) =A e^{-3 t}+B Y(t) =Ae−3t+B

Question 11

Consider this second-order nonhomogeneous differential equation:

Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)    y(t) =C1e2t(sin⁡(4t) +cos⁡(4t) ) +C2 y(t) =C_{1} e^{2 t}(\sin (4 t) +\cos (4 t) ) +C_{2} y(t) =C1​e2t(sin(4t) +cos(4t) ) +C2​ 
B)    y(t) =C1e4tsin⁡(2t) +C2e4tcos⁡(2t)  y(t) =C_{1} e^{4 t} \sin (2 t) +C_{2} e^{4 t} \cos (2 t)  y(t) =C1​e4tsin(2t) +C2​e4tcos(2t)  
C)    y(t) =C1e4t(sin⁡(2t) +cos⁡(2t) ) +C2 y(t) =C_{1} e^{4 t}(\sin (2 t) +\cos (2 t) ) +C_{2} y(t) =C1​e4t(sin(2t) +cos(2t) ) +C2​ 
D)    y(t) =C1e2tsin⁡(4t) +C2e2tcos⁡(4t)  y(t) =C_{1} e^{2 t} \sin (4 t) +C_{2} e^{2 t} \cos (4 t)  y(t) =C1​e2tsin(4t) +C2​e2tcos(4t)

Question 12

Suppose a 96-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 5 lbs for every foot per second of velocity. The form of the equation of unforced motion of the object in such a spring-mass system is  where m is the mass of the object, c is the damping constant, and k is the spring constant. Assume that the spring starts at a height of 3/2 feet below the horizontal with an upward velocity of - 2 feet per second.
For what values of A, B, C, and D can the solution,  be expressed in the form . Provide the exact values, not decimal approximations.
A = ____________, B = ____________, C = ____________, tan D = ____________

Accepted Answer

The answer of Suppose a 96-lb object stretches a spring...

Question 13

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

Accepted Answer

A)    (10  + 1)  = 0
B)   10  + 1y = 0
C)    (10  + 1y)  = 0
D)   10  + 1  = 0
E)   10  + 1y = 0

Question 14

Which of these is the general solution of the second-order nonhomogeneous differential equation  
and all capital letters are arbitrary real constants.

Accepted Answer

A)    y(t) =e3t(C1cos⁡(2t) +C2sin⁡(2t) ) +(At+
B)  e3tcos⁡(2t) +(Ct+
D)  e3tsin⁡(2t) +(Et2+Ft+G)  y(t) =e^{3 t}\left(C_{1} \cos (2 t) +C_{2} \sin (2 t) \right) +(A t+
B)   e^{3 t} \cos (2 t) +(C t+
D)   e^{3 t} \sin (2 t) +\left(E t^{2}+F t+G\right)  y(t) =e3t(C1​cos(2t) +C2​sin(2t) ) +(At+
B)  e3tcos(2t) +(Ct+
D)  e3tsin(2t) +(Et2+Ft+G)  
B)    y(t) =e3t(C1cos⁡(2t) +C2sin⁡(2t) ) +Ae3tcos⁡(2t) +Be3tsin⁡(2t) +Ct2 y(t) =e^{3 t}\left(C_{1} \cos (2 t) +C_{2} \sin (2 t) \right) +A e^{3 t} \cos (2 t) +B e^{3 t} \sin (2 t) +C t^{2} y(t) =e3t(C1​cos(2t) +C2​sin(2t) ) +Ae3tcos(2t) +Be3tsin(2t) +Ct2 
C)    y(t) =e3t(C1cos⁡(2t) +C2sin⁡(2t) ) +(At+
B)  e3tcos⁡(2t) +(Ct+
D)  e3tsin⁡(2t) +Et2 y(t) =e^{3 t}\left(C_{1} \cos (2 t) +C_{2} \sin (2 t) \right) +(A t+
B)   e^{3 t} \cos (2 t) +(C t+
D)   e^{3 t} \sin (2 t) +E t^{2} y(t) =e3t(C1​cos(2t) +C2​sin(2t) ) +(At+
B)  e3tcos(2t) +(Ct+
D)  e3tsin(2t) +Et2 
D)    y(t) =e2t(C1cos⁡(3t) +C2sin⁡(3t) ) +(At+
B)  e2tcos⁡(3t) +(Ct+
D)  e2tsin⁡(3t) +(Et2+Ft+G)  y(t) =e^{2 t}\left(C_{1} \cos (3 t) +C_{2} \sin (3 t) \right) +(A t+
B)   e^{2 t} \cos (3 t) +(C t+
D)   e^{2 t} \sin (3 t) +\left(E t^{2}+F t+G\right)  y(t) =e2t(C1​cos(3t) +C2​sin(3t) ) +(At+
B)  e2tcos(3t) +(Ct+
D)  e2tsin(3t) +(Et2+Ft+G)  
E)    y(t) =e2t(C1cos⁡(3t) +C2sin⁡(3t) ) +(At+
B)  e2tcos⁡(3t) +(Ct+
D)  e2tsin⁡(3t) +Et2 y(t) =e^{2 t}\left(C_{1} \cos (3 t) +C_{2} \sin (3 t) \right) +(A t+
B)   e^{2 t} \cos (3 t) +(C t+
D)   e^{2 t} \sin (3 t) +E t^{2} y(t) =e2t(C1​cos(3t) +C2​sin(3t) ) +(At+
B)  e2tcos(3t) +(Ct+
D)  e2tsin(3t) +Et2

Question 15

Consider this initial value problem:
  
Find a particular solution of the given nonhomogeneous differential equation.

Accepted Answer

The answer of Consider this initial value problem:
  
Find...

Question 16

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Accepted Answer

The answer of Suppose a 10-lb object stretches a spring...

Question 17

Consider this initial value problem:

For what values of $\alpha$ does the solution tend to 0 as t $\rightarrow$ $\infty$?

Accepted Answer

A)   all real numbers
B)   all nonzero real numbers
C)   all positive real numbers
D)   all negative real numbers

Question 18

Suppose a 160-lb object stretches a spring 12 feet while in equilibrium, and a dashpot provides a damping force of 5 lbs for every foot per second of velocity. The form of the equation of unforced motion of the object in such a spring-mass system is  where m is the mass of the object, c is the damping constant, and k is the spring constant. Assume that the spring starts at a height of 1/2 feet below the horizontal with an upward velocity of - 3/2 feet per second.
What is the quasi-frequency? Provide the exact values, not a decimal approximation.

Accepted Answer

The answer of Suppose a 160-lb object stretches a spring...

Question 19

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

Accepted Answer

A)    y=C1e4tsin⁡(6t) +C2e4tcos⁡(6t)  y=C_{1} e^{4 t} \sin (6 t) +C_{2} e^{4 t} \cos (6 t)  y=C1​e4tsin(6t) +C2​e4tcos(6t)  
B)    y=e−4t(C1sin⁡(6t) +C2cos⁡(6t) )  y=e^{-4 t}\left(C_{1} \sin (6 t) +C_{2} \cos (6 t) \right)  y=e−4t(C1​sin(6t) +C2​cos(6t) )  
C)    y=C1e4tcos⁡(6t) +C2e4tsin⁡(6t) +C y=C_{1} e^{4 t} \cos (6 t) +C_{2} e^{4 t} \sin (6 t) +C y=C1​e4tcos(6t) +C2​e4tsin(6t) +C 
D)    y=e6t(sin⁡(4t) +cos⁡(6t) ) +C y=e^{6 t}(\sin (4 t) +\cos (6 t) ) +C y=e6t(sin(4t) +cos(6t) ) +C 
E)    y=C1e−4tsin⁡(6t) +C2e−4tcos⁡(6t) +C y=C_{1} e^{-4 t} \sin (6 t) +C_{2} e^{-4 t} \cos (6 t) +C y=C1​e−4tsin(6t) +C2​e−4tcos(6t) +C

Question 20

Suppose a 128-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 5 lbs for every foot per second of velocity. The form of the equation of unforced motion of the object in such a spring-mass system is where m is the mass of the object, c is the damping constant, and k is the spring constant. Assume that the spring starts at a height of 5/2 feet below the horizontal with an upward velocity of - 3 feet per second. For what values of the arbitrary constants C₁ and C₂ does a general solution of the form y(t) = satisfy the initial conditions? Provide the exact values, not decimal approximations. C₁__________, C₂____________

Accepted Answer

The answer of Suppose a 128-lb object stretches a spring...

Exam 3: Second-Order Linear Differential Equations

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

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

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second. What is the natural frequency? Round your answer to the nearest hundredth radian per second.

For which of these differential equations is the characteristic equation given by 6 + 7 = 0?

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u₁ (t) and u₂ (t). u₁ (t) = u₂(t) =

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

Consider this initial value problem: Find a particular solution of the given nonhomogeneous differential equation.

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Exam 3: Second-Order Linear Differential Equations

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

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

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second. What is the natural frequency? Round your answer to the nearest hundredth radian per second.

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

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.

For which of these differential equations is the characteristic equation given by 6 + 7 = 0?

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u1 (t) and u2 (t). u1 (t) = ________ u2(t) = ________

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

What is the general solution of the homogeneous second-order Cauchy Euler differential equation are arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of these is a suitable form of a particular solution Y(t) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used? Here, all capital letters represent arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

Which of these is the general solution of the second-order nonhomogeneous differential equation and all capital letters are arbitrary real constants.

Consider this initial value problem: Find a particular solution of the given nonhomogeneous differential equation.

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Consider this initial value problem: For what values of α\alphaα does the solution tend to 0 as t →\rightarrow→ ∞\infty∞ ?

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

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

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u₁ (t) and u₂ (t). u₁ (t) = u₂(t) =