# Quiz 7: Separable Differential Equations; Integration Techniques

## Question 1

Which of the following differential equations are separable?

(i)
$\frac{dy}{dx}=xy$
(ii)
$\frac{dy}{dx}=x+y$
(iii)
$\frac{dy}{dx}=xy+y$

**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
$$\frac{dy}{dx}=3{x}^{2}y+2y.$$

**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 $\frac{dy}{dx}=\frac{-x}{y}$
are

**Not correct. Choice (a) is false.**

Try again. Note that
$\int y\phantom{\rule{0.3em}{0ex}}dy=-\int x\phantom{\rule{0.3em}{0ex}}dx$.

**Your answer is correct.**

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
$$\frac{dW}{dt}=0.05W-200.$$

**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
$\left({t}^{2}+1\right)\frac{dP}{dt}=Pt$, for
which $P\left(0\right)=3$.

**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 $\int \frac{1}{{sin}^{2}\theta \phantom{\rule{0.3em}{0ex}}d\theta}$.
(In each case, $C$
is an arbitrary constant.)

(Hint: $\frac{1}{{sin}^{2}\theta}=\frac{{sec}^{2}\theta}{{tan}^{2}\theta}$.)

(Hint: $\frac{1}{{sin}^{2}\theta}=\frac{{sec}^{2}\theta}{{tan}^{2}\theta}$.)

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

**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 ${\int}_{0}^{2\sqrt{3}}\frac{{x}^{2}}{\sqrt{16-{x}^{2}}}\phantom{\rule{0.3em}{0ex}}dx.$

**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 ${\int}_{1}^{2}\frac{1}{{x}^{2}\sqrt{{x}^{2}+1}}\phantom{\rule{0.3em}{0ex}}dx.$

**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 $\int \sqrt{4{x}^{2}-1}\phantom{\rule{0.3em}{0ex}}dx$.
More than one of the options may be correct.

(In each case, $C$ is an arbitrary constant.)

(In each case, $C$ is an arbitrary constant.)

**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**

**False**.**True**.**False**.**True**.