Classify the following differential equation:
$${e}^{x}\frac{dy}{dx}+3y={x}^{2}y$$
Exactly one option must be correct)

*Choice (a) is incorrect*

Note that
$\frac{dy}{dx}+{e}^{-x}\left(3-{x}^{2}\right)y=0$.

*Choice (b) is incorrect*

Note that $\frac{1}{y}\frac{dy}{dx}={e}^{-x}\left({x}^{2}-3\right)$.

*Choice (c) is correct!*

The equation can be written as
$\frac{1}{y}\frac{dy}{dx}={e}^{-x}\left({x}^{2}-3\right)$,
which shows that it is separable. It can also be written as
$\frac{dy}{dx}+{e}^{-x}\left(3-{x}^{2}\right)y=0$,
which shows that it is linear.

*Choice (d) is incorrect*

Note that
$\frac{1}{y}\frac{dy}{dx}={e}^{-x}\left({x}^{2}-3\right)$, and also
that $\frac{dy}{dx}+{e}^{-x}\left(3-{x}^{2}\right)y=0$.

Classify the following differential equation:
$$w\frac{dw}{dt}+3t=10$$
Exactly one option must be correct)

*Choice (a) is correct!*

Writing the equation
as $w\frac{dw}{dt}=10-3t$
shows that it is separable.

*Choice (b) is incorrect*

Note that
$w$ is multiplied
by $\frac{dw}{dt}$,
so the equation is not linear.

*Choice (c) is incorrect*

Note that
$w$ is multiplied
by $\frac{dw}{dt}$,
so the equation is not linear.

*Choice (d) is incorrect*

Note that
$w\frac{dw}{dt}=10-3t$,
which shows that the equation is separable.

Classify the following differential equation:
$$\frac{dx}{dt}=\frac{x+2xt+cost}{1+{t}^{2}}$$
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

In standard linear form the equation is
$\frac{dx}{dt}-\left(\frac{1+2t}{1+{t}^{2}}\right)x=\frac{cost}{1+{t}^{2}}.$

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Classify the following differential equation:
$$\frac{dz}{dt}=1+z+t+zt$$
Exactly one option must be correct)

*Choice (a) is incorrect*

Note that
$\frac{dz}{dt}-\left(1+t\right)z=1+t$.

*Choice (b) is incorrect*

Note that $\frac{dz}{dt}=\left(1+z\right)\left(1+t\right).$

*Choice (c) is incorrect*

Note that $\frac{dz}{dt}=\left(1+z\right)\left(1+t\right)$, and
also that $\frac{dz}{dt}-\left(1+t\right)z=1+t$.

*Choice (d) is correct!*

Writing $\frac{dz}{dt}=\left(1+z\right)\left(1+t\right)$ shows that
it is separable, and writing $\frac{dz}{dt}-\left(1+t\right)z=1+t$
shows that it is linear.

Suppose $y$ is a function of
$x$. Which of the following
is $\frac{d\left({x}^{3}y\right)}{dx}$? Exactly one option
must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

You must use the product rule for differentiation.

*Choice (c) is incorrect*

You must use the product rule for differentiation.

*Choice (d) is incorrect*

The derivative
of $y$ with
respect to $x$
is $\frac{dy}{dx}$.

Identify the functions $p\left(t\right)$
and $q\left(t\right)$ if the differential
equation $\phantom{\rule{1em}{0ex}}\frac{dx}{dt}=\frac{x+{t}^{2}-2x\sqrt{t}}{t}\phantom{\rule{1em}{0ex}}$ is written
in the form $\frac{dx}{dt}+p\left(t\right)x=q\left(t\right)$.
Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

An integrating factor, $I\left(x\right)$,
is found for the linear differential equation
$\left(1+{x}^{2}\right)\frac{dy}{dx}+xy=0,$
and the equation is rewritten as $\frac{d}{dx}\left(I\left(x\right)y\right)=0$.
Which of the following options is correct? Exactly one option must be correct)

*Choice (a) is incorrect*

Divide the
equation by $\left(1+{x}^{2}\right)$
first.

*Choice (b) is correct!*

$I\left(x\right)=\text{exp}\phantom{\rule{0.3em}{0ex}}\left(\int \frac{x}{1+{x}^{2}}\phantom{\rule{0.3em}{0ex}}dx\right)=\sqrt{1+{x}^{2}}.$

*Choice (c) is incorrect*

Try again.

*Choice (d) is incorrect*

Try
again.

Which of the following differential equations are equivalent to
$\phantom{\rule{1em}{0ex}}\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)={e}^{x}x$?

(More than one may be correct.) (Zero or more options can be correct)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be False.

For example, choice (d) should be True.

(More than one may be correct.) (Zero or more options can be correct)

*There is at least one mistake.*

For example, choice (a) should be False.

$\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.

*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.

$\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.

*There is at least one mistake.*

For example, choice (d) should be True.

*Correct!*

*False*$\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.*True**False*$\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.*True*

Consider the linear differential equation
$\phantom{\rule{1em}{0ex}}\frac{dy}{dx}+\frac{x}{1+x}y=1+x$.

The integrating factor is Exactly one option must be correct)

The integrating factor is Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is correct!*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Find the general solution to the differential equation
$\frac{dy}{dx}+\frac{x}{1+x}y=1+x$.

(In each of the following options $C$ is an arbitrary constant.) Exactly one option must be correct)

(In each of the following options $C$ is an arbitrary constant.) Exactly one option must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Remember to multiply the right hand side by the integrating factor.