Quiz 11: Second order linear differential equations with constant coefficients

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Question 1

What is the auxiliary equation associated with the second order linear differential equation
d2x dt2 + 19dx dt - 27x = 0?
a)
d2y dx2 + 19dy dx - 27y = 0
  b)
d2y dt2 + 19dy dt - 27y = 0
c)
k2 + 19k - 27 = 0
  d)
k2 - 19k + 27 = 0

 

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

Question 2

Find the general solution of the second order linear differential equation
d2y dx2 -dy dx - 12y = 0.
a)
y = Ae4x + Be-3x
  b)
y = Ae-4x + Be3x
c)
y = Ae6x + Be-2x
  d)
y = Ae-6x + Be2x

 

Your answer is correct.
k2 - k - 12 = (k - 4)(k + 3) = 0 hence k = 4 or k = -3 and the solution is y = Ae4x + Be-3x.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

Find the general solution to the second order linear differential equation
2d2x dt2 - 8dx dt + 6x = 0.
a)
x = Ae(-4+10)t + Be(-4-10)t
  b)
x = Ae(4+10)t + Be(4-10)t
c)
x = Ae3t + Bet
  d)
x = Ae-3t + Be-t

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
2k2 - 8k + 6 = 2(k - 3)(k - 1) = 0 hence k = 3 or k = 1 and the solution is x = Ae3t + Bet.
Not correct. Choice (d) is false.

Question 4

Find the particular solution to the second order linear differential equation
d2y dt2 + 6dy dt + 8y = 0,
where y = -2 and dy dt = 10 when t = 0.
a)
y = Ae-2t + Be-4t
  b)
y = Ae2t + Be4t
c)
y = -9e2t + 7e4t
  d)
y = e-2t - 3e-4t

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
k2 + 6k + 8 = (k + 2)(k + 4) = 0 hence k = -2 or k = -4 and the general solution is y = Ae-2t + Be-4t. Also dy dt = -2Ae-2t - 4Be-4t. At t = 0 we have A + B = -2 and - 2A - 4B = 10 which gives A = 1 and B = -3.

Question 5

Find the general solution to the second order linear differential equation
d2y dx2 - 14dy dx + 51y = 0.
a)
y = e7x(Asin2x + Bcos2x)
  b)
y = e2x(Asin7x + Bcos7x)
c)
y = Ae3x + Be-17x
  d)
y = Ae-3x + Be17x

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
k2 - 14k - 51 = (k - 17)(k + 3) = 0 hence k = 17 or k = -3 and the general solution is y = Ae-3x + Be17x.

Question 6

Find the particular solution to the second order linear differential equation
d2y dx2 - 6dy dx + 4y = 0
where y = 6 and dy dx = 18 when x = 0.
a)
y = 3e(3+5)x + 3e(3-5)x
  b)
y = 6e3x cos5x.
c)
y = e5x((6 - 25)sin3x + 6cos3x).
  d)
The initial conditions are inconsistent since the general solution is
y = Ae(-3+5)x + Be(-3-5)x.

 

Your answer is correct.
k2 - 6k - 16 = 0 has solutions k = 3 ±5 hence the general solution is y = Ae(3+5)x + Be(3-5)x. Since y = 6 when x = 0 we have A + B = 6. Now
dy dx = (3 + 5)Ae(3+5)x + (3 -5)Be(3-5)x
so we have 3(A + B) + 5(A - B) = 18 which gives A = B = 3 and y = 3e(3+5)x + 3e(3-5)x.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 7

Find the general solution to the second order linear differential equation
d2y dx2 - 6dy dx + 13y = 0.
a)
y = e2x(Asin3x + Bcos3x)
  b)
y = e3x(Asin2x + Bcos2x)
c)
y = e-2x(Asin3x + Bcos3x)
  d)
y = Aex + Be5x

 

Not correct. Choice (a) is false.
Your answer is correct.
k2 - 6k + 13 = 0 has solutions k = 3 ± 2-1 hence y = e3x(Asin2x + Bcos2x).
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 8

Find the particular solution of the second order linear differential equation
d2y dt2 - 8dy dt + 25y = 0
where y = -3 and dy dt = -6 when t = 0.
a)
y = e4t(2sin3t - 3cos3t)
  b)
y = e3t(2sin4t - 3cos4t)
c)
y = -e-4t(2sin3t + 3cos3t)
  d)
y = -e-3t(2sin4t + 3cos4t)

 

Your answer is correct.
k2 - 8k + 25 = 0 has solutions k = 4 ± 3-1 hence y = e4t(Asin3t + Bcos3t). At t = 0 we have y = B = -3 and
dy dt = 4e4t(Asin3t + Bcos3t) + e4t(3Acos3t - 3Bsin3t)
hence - 12 + 3A = -6 and A = 2. The particular solution is thus
y = e4t(2sin3t - 3cos3t).
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 9

Find the general solution to the second order linear differential equation
d2y dt2 - 25y = 0.
a)
y = Acos5t + Bsin5t
  b)
y = Ae-5t + Be5t
c)
y = 5cost + 5sint
  d)
y = A + Be25t

 

Not correct. Choice (a) is false.
Your answer is correct.
k2 - 25 = (k - 5)(k + 5) = 0 hence k = 5 or k = -5 and the general solution is y = Ae-5t + Be5t.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 10

Find the particular solution to the second order linear differential equation
d2x dt2 - 6dx dt + 8x = 0,
where x = -3 and dx dt = -6 when t = 0.
a)
x = Ae2t + Be4t
  b)
x = Ae-2t + Be-4t
c)
x = -3 2e2t -3 2e4t
  d)
x = -3e2t

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
k2 - 6k + 8 = 0 has solutions k = 2, k = 4 hence the general solution is x = Ae2t + Be4t. Since x = -3 when t = 0 we have A + B = -3
dx dt = 2Ae2t + 4Be4t
so we have 2A + 4B = -6 which gives A = -3 and B = 0. Hence x = -3e2t.
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