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Quiz 7:Separable Differential Equations; Integration Techniques

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following differential equations are separable?
(i) dy- dx = xy   (ii) dy- dx = x + y    (iii) dy- dx = xy + y
a) All three are separable.   b) Equation (i) only.
c) Equations (i) and (iii) only.   d) Equation (ii) only.

 

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 differential equation
dy-= 3x2y+ 2y. dx
a) y = Cex3+2x , where C is an arbitrary constant.
b) y = x3 + 2x + C, where C is an arbitrary constant.
c) y = ex3+2x + C , where C is an arbitrary constant.
d) y = ħex3+2x + C , where C is an arbitrary constant.

 

Your answer is correct.
Not correct. Choice (b) is false.
Always check your solution to a differential equation by differentiating.
Not correct. Choice (c) is false.
Always check your solution to a differential equation by differentiating.
Not correct. Choice (d) is false.
Always check your solution to a differential equation by differentiating.
 

Question 3

 
 
The graphs of the solutions to dy- dx = --x y are
a) straight lines;   b) circles;
c) parabolas;   d) hyperbolas.

 

Not correct. Choice (a) is false.
Try again. Note that ∫ ∫ ydy = - x dx .
Your answer is correct.
The solutions are x2 + y2 = C.
Not correct. Choice (c) is false.
Try again. Note that ∫ ∫ ydy = - x dx .
Not correct. Choice (d) is false.
Try again. Note that ∫ ∫ ydy = - xdx .
 

Question 4

 
 
Find the general solution of the differential equation
dW--= 0.05W - 200. dt
a) W = Aet + 4000, A an arbitrary constant.   b) W = Ae-10t, A an arbitrary constant.
c) W = e0.05t 0.05 + 4000 + A, A an arbitrary constant.   d) W = Ae0.05t + 4000, A an arbitrary constant.

 

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

Question 5

 
 
Find the particular solution of the differential equation (t2 + 1)dP- dt = Pt, for which P(0) = 3.
a) P = ln(t2 + 1) + 3.   b) P = 3∘ 2---- t+ 1.
c) P = 2 t- 2 + 3.   d) P = ∘ ----- t2 + 1 + 2.

 

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

Question 6

 
 
Find ∫ ---1--- sin2θ dθ .    (In each case, C is an arbitrary constant.)
(Hint: --1-- sin2 θ = 2 sec-θ- tan2 θ.)
a) --1-- tan θ + C   b) ln(sin2θ) + C
c) ln(tan2θ) + C   d) ---- 1--+ C sin θcosθ

 

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

Question 7

 
 
When the substitution x = 5sinht is made in the indefinite integral ∫ √--dx---- x2 + 25 , the result is:
a) ∫ ---1-- 5 cosh t dt   b) ∫ dt
c) ∫ 1 5 dt   d) None of the above.

 

Not correct. Choice (a) is false.
Don’t forget to substitute for dx.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 8

 
 
Evaluate the definite integral ∫ 2√3 √--x2----dx. 0 16- x2
a) 12   b) √ - --3 2
c) 8π- 3 - 2√3   d) 16√ - 3 + 4sin(4√ - 3)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Don’t forget to change the limits when you substitute.
 

Question 9

 
 
Evaluate the definite integral ∫ 2 --√-1----dx. 1 x2 x2 + 1
a) --1- sin1 --1-- sin2   b) -1-- sin1 + sin1 ---1- sin2 - sin2
c) 15√2 - 9√5- ----------- 10   d) 2√2-- √5- --------- 2

 

Not correct. Choice (a) is false.
Remember to change the limits if you make a substitution.
Not correct. Choice (b) is false.
Remember to substitute for dx, and change the limits if you make a substitution.
Not correct. Choice (c) is false.
Remember to substitute for dx when using substitution.
Your answer is correct.
 

Question 10

 
 
Find the indefinite integral ∫ ∘ ------- 4x2 - 1dx . More than one of the options may be correct.
(In each case, C is an arbitrary constant.)
a) x2-x+C   b) 1(2x√4x2---1- ln(2x + √4x2 --1))+ C 4
c) 2 2 (4x--- 1) 12x+ C   d) √ --2---- -1 x--4x---1- cosh--(2x)+ C 2 4

 

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