# Quiz 7: Separable Differential Equations; Integration Techniques

Question

## Question 1

Which of the following differential equations are separable?
(i) $\frac{dy}{dx}=xy$   (ii) $\frac{dy}{dx}=x+y$    (iii) $\frac{dy}{dx}=xy+y$
 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.
Not correct. Choice (d) is false.

## Question 2

Find the general solution of the differential equation $\frac{dy}{dx}=3{x}^{2}y+2y.$
 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.

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 $\frac{dy}{dx}=\frac{-x}{y}$ are
 a) straight lines; b) circles; c) parabolas; d) hyperbolas.

Not correct. Choice (a) is false.
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.
The solutions are ${x}^{2}+{y}^{2}=C$.
Not correct. Choice (c) is false.
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.
Not correct. Choice (d) is false.
Try again. Note that $\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.

## Question 4

Find the general solution of the differential equation $\frac{dW}{dt}=0.05W-200.$
 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.

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.

## Question 5

Find the particular solution of the differential equation $\left({t}^{2}+1\right)\frac{dP}{dt}=Pt$, for which $P\left(0\right)=3$.
 a) $P=ln\left({t}^{2}+1\right)+3$. b) $P=3\sqrt{{t}^{2}+1}$. c) $P=\frac{{t}^{2}}{2}+3$. d) $P=\sqrt{{t}^{2}+1}+2$.

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

## Question 6

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

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 $\int \frac{dx}{\sqrt{{x}^{2}+25}}$, the result is:
 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.

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

## Question 8

Evaluate the definite integral ${\int }_{0}^{2\sqrt{3}}\frac{{x}^{2}}{\sqrt{16-{x}^{2}}}\phantom{\rule{0.3em}{0ex}}dx.$
 a) $12$ b) $\frac{\sqrt{3}}{2}$ c) $\frac{8\pi }{3}-2\sqrt{3}$ d) $16\sqrt{3}+4sin\left(4\sqrt{3}\right)$

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

## Question 9

Evaluate the definite integral ${\int }_{1}^{2}\frac{1}{{x}^{2}\sqrt{{x}^{2}+1}}\phantom{\rule{0.3em}{0ex}}dx.$
 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) $\frac{2\sqrt{2}-\sqrt{5}}{2}$

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.

## Question 10

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.)
 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.