Quiz 11: Second Order Linear Differential Equations

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Question 1

If y = e2t is a solution to d2y dt2 - 5dy dt + ky = 0, what is the value of k?

 

Your answer is correct
Not correct. You may try again.

Question 2

If y = e3x cosx is a solution to d2y dx2 - 6dy dx + ky = 0, what is the value of k?

 

Your answer is correct
Not correct. You may try again.

Question 3

Which of the following is the general solution to d2y dx2 + 3dy dx - 10y = 0?
In each case, A and B are arbitrary constants.
a)
y = Ae2x + Be-5x
  b)
y = Ae-2x + Be5x
c)
y = Ae2x + Be5x
  d)
y = Ae-2x + Be-5x

 

Your answer is correct.
Not correct. Choice (b) is false.
Note the roots of the auxiliary equation are 2 and -5.
Not correct. Choice (c) is false.
Note the roots of the auxiliary equation are 2 and -5.
Not correct. Choice (d) is false.
Note that the roots of the auxiliary equation are 2 and -5.

Question 4

Which of the following is the general solution to d2y dt2 + 4dy dt - 4y = 0?
In each case, A and B are arbitrary constants.
a)
y = Ae-2t + Bte-2t
  b)
y = Ae2t + Be-2t
c)
y = Ae2t + Be-6t
  d)
y = Ae(-2+22)t + Be(-2-22)t

 

Not correct. Choice (a) is false.
Note that the roots of the auxiliary equation are -2 ± 22.
Not correct. Choice (b) is false.
Note that the roots of the auxiliary equation are -2 ± 22.
Not correct. Choice (c) is false.
Note that the roots of the auxiliary equation are -2 ± 22.
Your answer is correct.

Question 5

Which of the following are general solutions to d2x dt2 - 4dx dt + 13x = 0?
In each case, A and B are arbitrary constants. More than one option may be correct.
a)
x = Ae5t + Be-t
  b)
x = Ae(2+3i)t + Be(2-3i)t
c)
x = e2x(Acos3x + Bsin3x)
  d)
x = e3x(Acos2x + Bsin2x)

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
Your answers are correct
  1. False.
  2. True.
  3. True.
  4. False.

Question 6

Which of the following is the general solution to d2y dx2 + 10dy dx + 25y = 0?
In each case, A and B are arbitrary constants.
a)
y = Ae-5x + Be-5x
  b)
y = Axe-5x + Be-5x
c)
y = Ae5x + Be5x
  d)
y = Axe5x + Be5x

 

Not correct. Choice (a) is false.
Note that Ae-5x + Be-5x = (A + B)e-5x, so that there is really only one arbitrary constant involved in this option.
Your answer is correct.
Not correct. Choice (c) is false.
Note that Ae5x + Be5x = (A + B)e5x, so that there is really only one arbitrary constant involved in this option.
Not correct. Choice (d) is false.
The auxiliary equation has a repeated root of -5.

Question 7

Consider the differential equation d2y dx2 - 7y = 0.
Which of the following options is correct?
a)
The roots of the auxiliary equation are 0 and 7.
b)
There is no auxiliary equation for a differential equation of this type.
c)
The auxiliary equation has a repeated root of 7.
d)
The roots of the auxiliary equation 7 and -7.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.

Question 8

The general solution to d2z dt2 + 6dz dt + 9z = 0 is
z = Ate-3t + Be-3t.
Which of the following options is correct?
a)
As t , z A for any value of B.
b)
The behaviour of z as t depends on the values of A and B.
c)
As t , z 0 for any values of A and B.
d)
As t , z for any values of A and B.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.

Question 9

Find the particular solution to d2y dt2 - 3dy dt - 4y = 0 satisfying y(0) = 0 and y(0) = 5.
a)
y = et - e-4t
  b)
y = e-t - e4t
c)
y = e-4t - et
  d)
y = e4t - e-t

 

Not correct. Choice (a) is false.
This function satisfies the initial conditions, but not the differential equation.
Not correct. Choice (b) is false.
This function satisfies the differential equation, and y(0) = 0, but y(0) = -5.
Not correct. Choice (c) is false.
This functionsatisfies neither the differential equation, nor the initial conditions.
Your answer is correct.

Question 10

Find a solution to d2x dt2 + dx dt - 2x = 0 which satisfies x(0) = 3 and does not tend to infinity (or minus infinity) as t .
a)
x = 3e-t
  b)
x = 4e-2t - et
c)
x = 3e-2t
  d)
x = 2e-2t + e-t

 

Not correct. Choice (a) is false.
Try again. This function does not satisfy the differential equation.
Not correct. Choice (b) is false.
Try again. This function tends to - as t .
Your answer is correct.
Not correct. Choice (d) is false.
Try again. This function does not satisfy the differential equation.
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