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Quiz 8: First order differential equations and slope fields

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following differential equations are first order differential equations ?
a) d2y dy 2 dt2 + tdt = ty = sin t
b) 2dx- x dt + x = sint
c) dy- 2 dx = 4x
d) d2y dx2 = 4y

 

There is at least one mistake.
For example, choice (a) should be false.
This is a second order differential equation.
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.
This is a second order differential equation.
Your answers are correct
  1. False. This is a second order differential equation.
  2. True.
  3. True.
  4. False. This is a second order differential equation.
 

Question 2

 
 
Consider the first order differential equation dy dt = 4y where y = 1000 at t = 0. What is the general solution to the differential equation ?
a) y = 1000e4t   b) y = t4 + C
c) y = Be4t   d) y = t4 + 1000

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Note that the first response is the particular solution to the differential equation with the given initial condition.
Not correct. Choice (d) is false.
 

Question 3

 
 
Suppose you know that Q = Cekt satisfies the differential equation dQ dt-= - 2Q . What does this tell you about the values of C and k?
a) C = -2 and k = 1.   b) C = -1 and k = 2.
c) C = -2 and we know nothing of the value of k.   d) k = -2 and we know nothing of the value of C.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
kt dQ- kt Q = Ce ⇒ dt = Ckekt = kCe = kQ .
Hence k = -2 but we know nothing of the value of C.
 

Question 4

 
 
For what values of λ is y = Axλ a solution of the differential equation
2 x2d-y + 2xdy-- 6y = 0? dx2 dx
a) λ = 2 and λ = -3.   b) λ = -2 and λ = 3.
c) λ = -1 + √- 7 and λ = -1 -√ - 7.   d) There is not enough information to tell.

 

Your answer is correct.
If y = Axλ then dy- dx = λAxλ-1 and 2 d-y dx2 = λ(λ - 1)Axλ-2 . Hence
x2d2y dx2 + 2xdy dx- - 6y = 0
⇒ x2λ(λ - 1)Axλ-2 + 2xλAxλ-1 - 6Axλ = 0
⇒ λ(λ - 1)Axλ + 2λAxλ - 6Axλ = 0
⇒ λ(λ - 1) + 2λ - 6 = 0
⇒ λ2 + λ - 6 = (λ + 3)(λ - 2) = 0
⇒ λ = -3 and λ = 2.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 5

 
 
Consider the following direction (slope) field.
PIC
Which differential equation corresponds to this direction field ?
a) dy- 2 dx = 1+ y   b) -dy= x dx
c) -dy dx = x - y   d) -dy= 4 - y dx

 

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

Question 6

 
 
Consider the following direction (slope) field.
PIC
Which differential equation corresponds to this direction field ?
a) dy- dx = sin x   b) -dy= - y dx
c) dy dx-= - cosx   d) -dy dx = x

 

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

Question 7

 
 
Which of the direction fields below correspond to the differential equation dy dt = 3 -y?
a) PIC   b) PIC
c) PIC   d) PIC

 

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

Question 8

 
 
Which if the direction fields below correspond to the differential equation dy = - t dt ?
a) PIC   b) PIC
c) PIC   d) PIC

 

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

Question 9

 
 
Consider the direction field below for the differential equation y′ = 1 + y2 with the four curves labeled I to IV.
PIC
Which is the graph of the solution curve passing through (2,4) ?
a) Curve I   b) Curve II
c) Curve III   d) Curve IV

 

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

Question 10

 
 
Consider the direction field for y′ = 1(1+ y)(2 - y) 2 below.
PIC
Which points could possibly lie on the same solution curve ?
a) (-1,-1), (0,0), (1,1), (1,2).   b) (-1,-1), (0,1 2), (1,1 2), (2,5).
c) (-1,-4), (0,-3), (1,-2), (2,1).   d) (-1,-1 - 2), (0,12), (1,12), (2,1).

 

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