## MATH1003 Quizzes

Quiz 9: First Order Linear Differential Equations
Question 1 Questions
Classify the following differential equation: ${e}^{x}\frac{dy}{dx}+3y={x}^{2}y$ Exactly one option must be correct)
 a) Separable and not linear. b) Linear and not separable. c) Both separable and linear. d) Neither separable nor linear.

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)
 a) Separable and not linear. b) Linear and not separable. c) Both separable and linear. d) Neither separable nor linear.

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)
 a) Separable and not linear. b) Linear and not separable. c) Both separable and linear. d) Neither separable nor linear.

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)
 a) Separable and not linear. b) Linear and not separable. c) Neither separable nor linear. d) Both separable and linear.

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)
 a) $3{x}^{2}y+{x}^{3}\frac{dy}{dx}$ b) $3{x}^{2}y$ c) $3{x}^{2}\frac{dy}{dx}$ d) $3{x}^{2}y+{x}^{3}$

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)
 a) It is not possible to write the equation in the form described. b) $p\left(t\right)=2\sqrt{t}-1$, $q\left(t\right)={t}^{2}$. c) $p\left(t\right)=1-2\sqrt{t}$, $q\left(t\right)=t$. d) $p\left(t\right)=\frac{2\sqrt{t}-1}{t}$, $q\left(t\right)=t$.

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)
 a) $I\left(x\right)={e}^{{x}^{2}∕2}$ b) $I\left(x\right)=\sqrt{1+{x}^{2}}$ c) $I\left(x\right)=1+{x}^{2}$ d) It is not possible to find such a function $I\left(x\right)$.

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)
 a) ${e}^{x}\frac{dy}{dx}={e}^{x}x$ b) ${e}^{x}\frac{dy}{dx}+{e}^{x}y={e}^{x}x$ c) $\frac{dy}{dx}=x$ d) $\frac{dy}{dx}=x-y$

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!
1. False $\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.
2. True
3. False $\frac{d\phantom{\rule{0.3em}{0ex}}}{dx}\left({e}^{x}y\right)\ne {e}^{x}\frac{dy}{dx}$.
4. 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)
 a) ${e}^{x}$ b) $\frac{{e}^{x}}{1+x}$ c) ${e}^{x}\left(1+x\right)$ d) ${e}^{x+{x}^{2}∕2}$

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)
 a) $\left(1+C{e}^{-x}\right)\left(1+x\right)$ b) $1+x+C$ c) $C\phantom{\rule{0.3em}{0ex}}\left(1+x\right)$ d) ${e}^{-x}\left(x+\frac{{x}^{2}}{2}+C\phantom{\rule{0.3em}{0ex}}\right)\left(1+x\right)$

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.