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Quiz 12: Systems of Differential Equations

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Question 1

Which of the following functions x(t) and y(t) satisfy the differential equations

dx dy ---= 2xt; y-- = xt, dt dt
with x(0) = 4 and y(0) = 1?
a)
2 √ -------- x = et+ 3; y = et2 + 3t2
b)
2 √------- x = 4et ; y = 4et2 - 3
c)
√ --- x = 4et2; y = 2 et2 - 1
d)
x = 4et2; y = 2et - 1

 

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

Question 2

If x(t) and y(t) satisfy the differential equations

′ ′ x = - 0.05y; y = - 0.01x,

and x′′ + ax = 0 is the second order differential equation satisfied by x(t) , what is the value of a?

 

Your answer is correct
Not correct. You may try again.
Note that x ′′ = - 0.05y′ .

Question 3

If x(t) and y(t) satisfy the differential equations

dx dy dt-= - 2x + 3y; dt = 7x - 5y,

which of the following is the second order differential equation satisfied by x(t) ?
a)
2 ddxt2-+ 7ddxt - 11x = 0
  b)
2 ddtx2 - 9 dxdt - 2x = 0
c)
2 ddtx2 + 7ddxt +5x = 0
  d)
d2dtx2 - 7 dxdt - 4x = 0

 

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

Question 4

If x(t) and y(t) satisfy the differential equations

x′ = 2x- 3y; y′ = - x+ y,

which of the following is the second order differential equation satisfied by y(t) ?
a)
y′′ - 3y′+ y = 0
  b)
y′′ + 3y′ + y = 0
c)
y′′ - 3y′ - y = 0
  d)
None of the above.

 

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

Question 5

Choose the option which is a general solution to the system of differential equations:

dx dy -dt = 4x - 3y; -dt = 2x - y.

In each case, A and B are arbitrary constants.
a)
x = Aet +Be2t; y = 5Aet + 6Be2t
b)
x = Aet + Be2t; y = Aet + Be2t
c)
x = Aet + Be2t; y = 3Aet +2Be2t
d)
x = Aet + Be2t; y = Aet + 23Be2t

 

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

Question 6

Choose the option which is a general solution to the system of differential equations:

dx- dy dt = - 2x+ y; dt = - x - 4y.

In each case, A and B are arbitrary constants.
a)
-3t -3t x = e (At +B ); y = e (A - At - B )
b)
-3t -3t x = e (At + B); y = e (At + B)
c)
t 7t t 7t x = Ae + Be ; y = Ae + Be
d)
t 7t t 7t x = Ae + Be ; y = 3Ae + 9Be

 

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

Question 7

Choose the options which are particular solutions to the system of differential equations:

dx dy dt = x + 5y; dt = - x- y.

More than one option my be correct.
a)
x = cos2t; y = - 25 sin 2t - 15 cos2t
b)
x = cos2t; y = sin2t+ cos2t
c)
x = sin 2t; y = - 15 sin2t+ 25 cos2t
d)
x = 5cos2t; y = - 2 sin2t- cos2t

 

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

Question 8

In a certain environment live 10000 particularly harmful insects of species X. In the hope of eradicating species X, 1000 insects of species Y , which eat species X, are introduced. Let X(t) be the number of thousands of insects of species X, and Y (t) be the number of thousands of insects of species Y , at time t (in years) after species Y is introduced. Suppose the sizes of the populations are governed by

X ′ = 5X - Y; Y′ = 5X + Y.

According to this model, which of the following options is correct?
a)
Both species survive indefinitely.
b)
Both species survive for approximately 2.7 years, at which time species X is eradicated.
c)
Both species survive for approximately 3.2 years, at which time species Y dies out.
d)
Both species survive for approximately 153 years, at which time species X is eradicated.

 

Not correct. Choice (a) is false.
The population sizes are given by X = e3t(10cost+ 19sin t), Y = e3t(cost+ 48sin t) .
Look for the smallest value of t that makes X or Y equal to zero.
Your answer is correct.
The population sizes are given by 3t 3t X = e (10cost+ 19sin t), Y = e (cost+ 48sin t) .
Not correct. Choice (c) is false.
The population sizes are given by X = e3t(10cost+ 19sint), Y = e3t(cost+ 48 sint) .
X is equal to zero before t = 3.2.
Not correct. Choice (d) is false.
The population sizes are given by X = e3t(10cost+ 19sin t), Y = e3t(cost+ 48sin t) .
Look for the smallest value of t in RADIANS that makes X or Y equal to zero.

Question 9

Questions 9 and 10 use the following model for a battle between two armies, X and Y . The model assumes that the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the number of soldiers in the opposing army. For one particular battle, the proposed model is

dX dY -dt = - 0.04Y; -dt = - 0.01X

where X(t) and Y (t) are the numbers of soldiers still fighting in armies X and Y , respectively, t days after the battle starts. The initial strength of army X was 54 000, and that of army Y was 21 500. If army Y fought without surrendering until all its soldiers were killed how long did the battle last? (Give your answer rounded up to the nearest day.)

 

Your answer is correct
Not correct. You may try again.
The particular solution for Y is Y = 24250e-0.02t - 2750e0.02t. Solving Y = 0 gives t = 55.

Question 10

Questions 9 and 10 use the following model for a battle between two armies, X and Y . The model assumes that the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the number of soldiers in the opposing army. For one particular battle, the proposed model is

dX- = - 0.04Y; dY- = - 0.01X dt dt

where X(t) and Y (t) are the numbers of soldiers still fighting in armies X and Y , respectively, t days after the battle starts. The initial strength of army X was 54 000, and that of army Y was 21 500.
Use the answer to question 9 to determine the approximate number of soldiers left in army X when the battle is over. (Give your answer to the nearest integer.)

 

Your answer is correct
Not correct. You may try again.
The particular solution for X is X = 5500e0.02t + 48500e-0.02t. Substitute t = 55.
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