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MATH1003 Quizzes

Quiz 7: Separable Differential Equations; Integration Techniques
Question 1 Questions
Which of the following differential equations are separable?
(i) dy dx = xy (ii) dy dx = x + y    (iii) dy dx = xy + y
Exactly one option must be correct)
a)
All three are separable.
b)
Equation (i) only.
c)
Equations (i) and (iii) only.
d)
Equation (ii) only.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Find the general solution of the differential equation dy dx = 3x2y + 2y. Exactly one option must be correct)
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 = ±ex3+2x + C, where C is an arbitrary constant.

Choice (a) is correct!
Choice (b) is incorrect
Always check your solution to a differential equation by differentiating.
Choice (c) is incorrect
Always check your solution to a differential equation by differentiating.
Choice (d) is incorrect
Always check your solution to a differential equation by differentiating.
The graphs of the solutions to dy dx = x y are Exactly one option must be correct)
a)
straight lines;
b)
circles;
c)
parabolas;
d)
hyperbolas.

Choice (a) is incorrect
Try again. Note that ydy = xdx.
Choice (b) is correct!
The solutions are x2 + y2 = C.
Choice (c) is incorrect
Try again. Note that ydy = xdx.
Choice (d) is incorrect
Try again. Note that ydy = xdx.
Find the general solution of the differential equation dW dt = 0.05W 200. Exactly one option must be correct)
a)
W = Aet + 4000, A an arbitrary constant.
b)
W = Ae10t, 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.

Choice (a) is incorrect
Check by differentiation.
Choice (b) is incorrect
Check by differentiation.
Choice (c) is incorrect
Check by differentiation.
Choice (d) is correct!
Find the particular solution of the differential equation (t2 + 1)dP dt = Pt, for which P(0) = 3. Exactly one option must be correct)
a)
P = ln(t2 + 1) + 3.
b)
P = 3t2 + 1.
c)
P = t2 2 + 3.
d)
P = t2 + 1 + 2.

Choice (a) is incorrect
Check by differentiation.
Choice (b) is correct!
Choice (c) is incorrect
Check by differentiation.
Choice (d) is incorrect
Check by differentiation.
Find 1 sin2θdθ. (In each case, C is an arbitrary constant.)
(Hint: 1 sin 2θ = sec 2θ tan 2θ.) Exactly one option must be correct)
a)
1 tan θ + C
b)
ln(sin2θ) + C
c)
ln(tan2θ) + C
d)
1 sinθcosθ + C

Choice (a) is correct!
Choice (b) is incorrect
Check by differentiation.
Choice (c) is incorrect
Check by differentiation.
Choice (d) is incorrect
Check by differentiation.
When the substitution x = 5sinht is made in the indefinite integral dx x2 + 25, the result is: Exactly one option must be correct)
a)
1 5coshtdt
b)
dt
c)
1 5dt
d)
None of the above.

Choice (a) is incorrect
Don’t forget to substitute for dx.
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Evaluate the definite integral 023 x2 16 x2dx. Exactly one option must be correct)
a)
12
b)
3 2
c)
8π 3 23
d)
163 + 4sin(43)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Don’t forget to change the limits when you substitute.
Evaluate the definite integral 12 1 x2x2 + 1dx. Exactly one option must be correct)
a)
1 sin 1 1 sin 2
b)
1 sin 1 + sin1 1 sin 2 sin2
c)
15295 10
d)
225 2

Choice (a) is incorrect
Remember to change the limits if you make a substitution.
Choice (b) is incorrect
Remember to substitute for dx, and change the limits if you make a substitution.
Choice (c) is incorrect
Remember to substitute for dx when using substitution.
Choice (d) is correct!
Find the indefinite integral 4x2 1dx. More than one of the options may be correct.
(In each case, C is an arbitrary constant.) (Zero or more options can be correct)
a)
x2 x + C
b)
1 4(2x4x2 1 ln(2x + 4x2 1)) + C
c)
(4x21)2 12x + C
d)
x4x2 1 2 cosh1(2x) 4 + C

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.
Correct!
  1. False
  2. True
  3. False
  4. True