MATH1003 Quizzes

Quiz 7: Separable Differential Equations; Integration Techniques
Question 1 Questions
Which of the following differential equations are separable?
(i) $\frac{dy}{dx}=xy$ (ii) $\frac{dy}{dx}=x+y$    (iii) $\frac{dy}{dx}=xy+y$
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 $\frac{dy}{dx}=3{x}^{2}y+2y.$ Exactly one option must be correct)
 a) $y=C{e}^{{x}^{3}+2x}$, where $C$ is an arbitrary constant. b) $y={x}^{3}+2x+C$, where $C$ is an arbitrary constant. c) $y={e}^{{x}^{3}+2x}+C$, where $C$ is an arbitrary constant. d) $y=±{e}^{{x}^{3}+2x}+C$, 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 $\frac{dy}{dx}=\frac{-x}{y}$ are Exactly one option must be correct)
 a) straight lines; b) circles; c) parabolas; d) hyperbolas.

Choice (a) is incorrect
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.
Choice (b) is correct!
The solutions are ${x}^{2}+{y}^{2}=C$.
Choice (c) is incorrect
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.
Choice (d) is incorrect
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.
Find the general solution of the differential equation $\frac{dW}{dt}=0.05W-200.$ Exactly one option must be correct)
 a) $W=A{e}^{t}+4000$, $A$ an arbitrary constant. b) $W=A{e}^{-10t}$, $A$ an arbitrary constant. c) $W=\frac{{e}^{0.05t}}{0.05}+4000+A$, $A$ an arbitrary constant. d) $W=A{e}^{0.05t}+4000$, $A$ 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 $\left({t}^{2}+1\right)\frac{dP}{dt}=Pt$, for which $P\left(0\right)=3$. Exactly one option must be correct)
 a) $P=ln\left({t}^{2}+1\right)+3$. b) $P=3\sqrt{{t}^{2}+1}$. c) $P=\frac{{t}^{2}}{2}+3$. d) $P=\sqrt{{t}^{2}+1}+2$.

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 $\int \frac{1}{{sin}^{2}\theta \phantom{\rule{0.3em}{0ex}}d\theta }$. (In each case, $C$ is an arbitrary constant.)
(Hint: $\frac{1}{{sin}^{2}\theta }=\frac{{sec}^{2}\theta }{{tan}^{2}\theta }$.) Exactly one option must be correct)
 a) $\frac{-1}{tan\theta }+C$ b) $ln\left({sin}^{2}\theta \right)+C$ c) $ln\left({tan}^{2}\theta \right)+C$ d) $\frac{-1}{sin\theta cos\theta }+C$

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 $\int \frac{dx}{\sqrt{{x}^{2}+25}}$, the result is: Exactly one option must be correct)
 a) $\int \frac{1}{5cosht}\phantom{\rule{0.3em}{0ex}}dt$ b) $\int \phantom{\rule{0.3em}{0ex}}dt$ c) $\int \frac{1}{5}\phantom{\rule{0.3em}{0ex}}dt$ 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 ${\int }_{0}^{2\sqrt{3}}\frac{{x}^{2}}{\sqrt{16-{x}^{2}}}\phantom{\rule{0.3em}{0ex}}dx.$ Exactly one option must be correct)
 a) $12$ b) $\frac{\sqrt{3}}{2}$ c) $\frac{8\pi }{3}-2\sqrt{3}$ d) $16\sqrt{3}+4sin\left(4\sqrt{3}\right)$

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 ${\int }_{1}^{2}\frac{1}{{x}^{2}\sqrt{{x}^{2}+1}}\phantom{\rule{0.3em}{0ex}}dx.$ Exactly one option must be correct)
 a) $\frac{1}{sin1}-\frac{1}{sin2}$ b) $\frac{1}{sin1}+sin1-\frac{1}{sin2}-sin2$ c) $\frac{15\sqrt{2}-9\sqrt{5}}{10}$ d) $\frac{2\sqrt{2}-\sqrt{5}}{2}$

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 $\int \sqrt{4{x}^{2}-1}\phantom{\rule{0.3em}{0ex}}dx$. 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) ${x}^{2}-x+C$ b) $\frac{1}{4}\left(2x\sqrt{4{x}^{2}-1}-ln\left(2x+\sqrt{4{x}^{2}-1}\right)\right)+C$ c) $\frac{{\left(4{x}^{2}-1\right)}^{2}}{12x}+C$ d) $\frac{x\sqrt{4{x}^{2}-1}}{2}-\frac{{cosh}^{-1}\left(2x\right)}{4}+C$

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