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Quiz 7: Separable Differential Equations; Integration Techniques

Question

Question 1

Which of the following differential equations are separable?

(i)

\frac{d y}{d x} = x y

(ii)

\frac{d y}{d x} = x + y

(iii)

\frac{d y}{d x} = x y + 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{d y}{d x} = 3 x^{2} y + 2 y .

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{d y}{d x} = \frac{- x}{y}

are

Not correct. Choice (a) is false.

Try again. Note that

\int y d y = - \int x d x

Your answer is correct.

The solutions are

x^{2} + y^{2} = C

Not correct. Choice (c) is false.

Try again. Note that

\int y d y = - \int x d x

Not correct. Choice (d) is false.

Try again. Note that

\int y d y = - \int x d x

Question 4

Find the general solution of the differential equation

\frac{d W}{d t} = 0.05 W - 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

(t^{2} + 1) \frac{d P}{d t} = P t

, for which

P (0) = 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} θ d θ}

. (In each case,

C

is an arbitrary constant.)
(Hint:

\frac{1}{{sin}^{2} θ} = \frac{{sec}^{2} θ}{{tan}^{2} θ}

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 = 5 sinh t

is made in the indefinite integral

\int \frac{d x}{\sqrt{x^{2} + 25}}

, the result is:

Not correct. Choice (a) is false.

Don’t forget to substitute for

d x

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}}} d x .

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}} d x .

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

d x

, and change the limits if you make a substitution.

Not correct. Choice (c) is false.

Remember to substitute for

d x

when using substitution.

Your answer is correct.

Question 10

Find the indefinite integral

\int \sqrt{4 x^{2} - 1} d x

. More than one of the options may be correct.
(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.

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Questions:

right first attempt
right
wrong

a)	All three are separable.	b)	Equation (i) only.
c)	Equations (i) and (iii) only.	d)	Equation (ii) only.

a)		$y = C e^{x^{3} + 2 x}$ , where $C$ is an arbitrary constant.
b)		$y = x^{3} + 2 x + C$ , where $C$ is an arbitrary constant.
c)		$y = e^{x^{3} + 2 x} + C$ , where $C$ is an arbitrary constant.
d)		$y = \pm e^{x^{3} + 2 x} + C$ , where $C$ is an arbitrary constant.

a)	$W = A e^{t} + 4000$ , $A$ an arbitrary constant.	b)	$W = A e^{- 10 t}$ , $A$ an arbitrary constant.
c)	$W = \frac{e^{0.05 t}}{0.05} + 4000 + A$ , $A$ an arbitrary constant.	d)	$W = A e^{0.05 t} + 4000$ , $A$ an arbitrary constant.

a)	$P = ln (t^{2} + 1) + 3$ .	b)	$P = 3 \sqrt{t^{2} + 1}$ .
c)	$P = \frac{t^{2}}{2} + 3$ .	d)	$P = \sqrt{t^{2} + 1} + 2$ .

a)	$\frac{- 1}{tan θ} + C$	b)	$ln ({sin}^{2} θ) + C$
c)	$ln ({tan}^{2} θ) + C$	d)	$\frac{- 1}{sin θ cos θ} + C$

a)	$\int \frac{1}{5 cosh t} d t$	b)	$\int d t$
c)	$\int \frac{1}{5} d t$	d)	None of the above.

a)	$12$	b)	$\frac{\sqrt{3}}{2}$
c)	$\frac{8 π}{3} - 2 \sqrt{3}$	d)	$16 \sqrt{3} + 4 sin (4 \sqrt{3})$

a)	$\frac{1}{sin 1} - \frac{1}{sin 2}$	b)	$\frac{1}{sin 1} + sin 1 - \frac{1}{sin 2} - sin 2$
c)	$\frac{15 \sqrt{2} - 9 \sqrt{5}}{10}$	d)	$\frac{2 \sqrt{2} - \sqrt{5}}{2}$

a)		$x^{2} - x + C$		b)		$\frac{1}{4} (2 x \sqrt{4 x^{2} - 1} - ln (2 x + \sqrt{4 x^{2} - 1})) + C$
c)		$\frac{{(4 x^{2} - 1)}^{2}}{12 x} + C$		d)		$\frac{x \sqrt{4 x^{2} - 1}}{2} - \frac{{cosh}^{- 1} (2 x)}{4} + C$

a)	straight lines;	b)	circles;
c)	parabolas;	d)	hyperbolas.

The University of Sydney - School of Mathematics and Statistics

Quiz 7: Separable Differential Equations; Integration Techniques

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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