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Quiz 9: First Order Linear Differential Equations

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Classify the following differential equation:
ex dy+ 3y = x2y dx
a) Separable and not linear.   b) Linear and not separable.
c) Both separable and linear.   d) Neither separable nor linear.

 

Not correct. Choice (a) is false.
Note that dy- dx + e-x(3 -x2)y = 0.
Not correct. Choice (b) is false.
Note that 1- ydy- dx = e-x(x2 - 3).
Your answer is correct.
The equation can be written as -1 ydy- dx = e-x(x2 - 3), which shows that it is separable. It can also be written as dy- dx + e-x(3 -x2)y = 0, which shows that it is linear.
Not correct. Choice (d) is false.
Note that 1 -- ydy --- dx = e-x(x2 - 3), and also that dy --- dx + e-x(3 - x2)y = 0.
 

Question 2

 
 
Classify the following differential equation:
dw w ---+ 3t = 10 dt
a) Separable and not linear.   b) Linear and not separable.
c) Both separable and linear.   d) Neither separable nor linear.

 

Your answer is correct.
Writing the equation as wdw- dt = 10 - 3t shows that it is separable.
Not correct. Choice (b) is false.
Note that w is multiplied by dw -dt, so the equation is not linear.
Not correct. Choice (c) is false.
Note that w is multiplied by dw- dt, so the equation is not linear.
Not correct. Choice (d) is false.
Note that wdw- dt = 10 - 3t, which shows that the equation is separable.
 

Question 3

 
 
Classify the following differential equation:
dx- x-+-2xt+-cos-t dt = 1+ t2
a) Separable and not linear.   b) Linear and not separable.
c) Both separable and linear.   d) Neither separable nor linear.

 

Not correct. Choice (a) is false.
Your answer is correct.
In standard linear form the equation is dx dt--(1 + 2t) 1+-t2x = cost 1-+-t2.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 4

 
 
Classify the following differential equation:
dz= 1 + z + t+ zt dt
a) Separable and not linear.   b) Linear and not separable.
c) Neither separable nor linear.   d) Both separable and linear.

 

Not correct. Choice (a) is false.
Note that dz dt - (1 + t)z = 1 + t.
Not correct. Choice (b) is false.
Note that dz dt = (1 + z)(1 + t).
Not correct. Choice (c) is false.
Note that dz -- dt = (1 + z)(1 + t), and also that dz -- dt - (1 + t)z = 1 + t.
Your answer is correct.
Writing dz -dt = (1 + z)(1 + t) shows that it is separable, and writing dz dt- (1 + t)z = 1 + t shows that it is linear.
 

Question 5

 
 
Suppose y is a function of x. Which of the following is d(x3y)- dx?
a) 3x2y + x3dy- dx   b) 3x2y
c) 3x2dy- dx   d) 3x2y + x3

 

Your answer is correct.
Not correct. Choice (b) is false.
You must use the product rule for differentiation.
Not correct. Choice (c) is false.
You must use the product rule for differentiation.
Not correct. Choice (d) is false.
The derivative of y with respect to x is dy- dx.
 

Question 6

 
 
Identify the functions p(t) and q(t) if the differential equation dx- dt = √- x+-t2 --2x-t t is written in the form dx- dt + p(t)x = q(t) .
a) It is not possible to write the equation in the form described.   b) √ - p(t) = 2 t- 1 , q(t) = t2 .
c) √ - p(t) = 1- 2 t , q(t) = t .   d) √- p(t) = 2-tt-1 , q(t) = t .

 

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

Question 7

 
 
An integrating factor, I(x) , is found for the linear differential equation (1 + x2)dy- dx + xy = 0, and the equation is rewritten as d-(I(x)y) = 0 dx . Which of the following options is correct?
a) I(x) = ex2∕2   b) I(x) = √1-+-x2
c) 2 I(x) = 1 + x   d) It is not possible to find such a function I(x) .

 

Not correct. Choice (a) is false.
Divide the equation by (1 + x2) first.
Your answer is correct.
I(x) = exp (∫--x-dx) = √1-+-x2. 1+x2
Not correct. Choice (c) is false.
Try again.
Not correct. Choice (d) is false.
Try again.
 

Question 8

 
 
Which of the following differential equations are equivalent to d-- dx(exy) = exx?
(More than one may be correct.)
a) exdy- dx = exx   b) exdy- dx + exy = exx
c) -dy dx = x   d) dy- dx = x - y

 

There is at least one mistake.
For example, choice (a) should be false.
-d(exy) ⁄= exdy dx dx .
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 false.
ddx(exy) ⁄= exddyx .
There is at least one mistake.
For example, choice (d) should be true.
Your answers are correct
  1. False. -d(exy) ⁄= exdy dx dx .
  2. True.
  3. False. ddx(exy) ⁄= exddyx .
  4. True.
 

Question 9

 
 
Consider the linear differential equation dy- dx + -x--- 1+ xy = 1 + x.
The integrating factor is
a) ex   b) x --e-- 1 + x
c) ex(1 +x)   d) ex+x2∕2

 

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

Question 10

 
 
Find the general solution to the differential equation dy- dx + --x-- 1 + xy = 1 + x.
(In each of the following options C is an arbitrary constant.)
a) (1+ Ce- x)(1+ x)   b) 1 + x+ C
c) C (1+ x)   d) e-x(x+ x2-+ C )(1+ x) 2

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Remember to multiply the right hand side by the integrating factor.