## MATH1013 Quizzes

Quiz 9: Separable differential equations
Question 1 Questions
Find the solution to the differential equation
$\frac{dy}{dx}=2y\phantom{\rule{0.3em}{0ex}}.$
 a) $y={x}^{2}+C$ b) $y={e}^{\frac{1}{2}x}$ c) $y=A{e}^{2x}$ d) $y=\frac{1}{2}A{e}^{x}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
$\begin{array}{llll}\hfill \frac{dy}{dx}& =2y,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dy}{y}& =\int 2dx,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|y|& =2x+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |y|& ={e}^{2x+C}=B{e}^{2x}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}B={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =A{e}^{2x}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (d) is incorrect
Find the solution to the differential equation
$\frac{dP}{dt}=3P\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}P=4\phantom{\rule{1em}{0ex}}when\phantom{\rule{1em}{0ex}}t=0\phantom{\rule{0.3em}{0ex}}.$
 a) $P=A{e}^{3t}$ b) $P=3A{e}^{t}$ c) $P=12A{e}^{t}$ d) $P=4{e}^{3t}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Find the solution to the differential equation
$\frac{dy}{dx}=xy\phantom{\rule{0.3em}{0ex}}.$
 a) $y=\frac{1}{2}{x}^{2}+C$ b) $y=B{e}^{\frac{1}{2}{x}^{2}}$ c) $y={e}^{\frac{1}{2}{x}^{2}}$ d) $y=\frac{1}{2}B{e}^{{x}^{2}}$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{llll}\hfill \frac{dy}{dx}& =xy,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dy}{y}& =\int x\phantom{\rule{0.3em}{0ex}}dx,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|y|& =\frac{1}{2}{x}^{2}+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |y|& ={e}^{\frac{1}{2}x+C}=A{e}^{\frac{1}{2}x}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}A={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =B{e}^{\frac{1}{2}x}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Find the solution to the differential equation
$\frac{dP}{dt}=4P+5\phantom{\rule{0.3em}{0ex}}.$
 a) $P=A{e}^{4t}+5t$ b) $P=A{e}^{4t}-1.25$ c) $P=5{e}^{4t}+5t$ d) $P=A{e}^{4t}+1.25t$

Choice (a) is incorrect
Choice (b) is correct!
$\begin{array}{llll}\hfill \frac{dP}{dt}& =4P+5=4\left(P+1.25\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dP}{\left(P+1.25\right)}& =\int 4\phantom{\rule{0.3em}{0ex}}dt,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|P+1.25|& =4t+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |P+1.25|& ={e}^{4t+C}=B{e}^{4t}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}B={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill P+1.25& =A{e}^{4t},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill P& =A{e}^{4t}-1.25.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Find the solution to the differential equation
$\frac{dP}{dt}=2P-2Pt\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}P=5\phantom{\rule{1em}{0ex}}when\phantom{\rule{1em}{0ex}}t=0\phantom{\rule{0.3em}{0ex}}.$
 a) $P=5{e}^{2-2t}$ b) $P=5{e}^{2t-{t}^{2}}$ c) $P=5{e}^{2-2{t}^{2}}$ d) $P={P}^{2}-{P}^{2}t$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Find the solution to the differential equation
$\frac{dN}{dt}-\frac{N}{k}=0\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}k\phantom{\rule{1em}{0ex}}is\phantom{\rule{1em}{0ex}}a\phantom{\rule{1em}{0ex}}constant.$
 a) $N=B{e}^{\frac{t}{k}}$ where $B$ is a constant. b) $N=B{e}^{-kt}$ where $B$ is a constant. c) $N=B{e}^{-\frac{t}{k}}$ where $B$ is a constant. d) $N=B{e}^{kt}$ where $B$ is a constant.

Choice (a) is correct!
$\begin{array}{llll}\hfill \frac{dN}{dt}& =\frac{N}{k},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dN}{N}& =\int \frac{1}{k},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|N|& =\frac{t}{k}+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |N|& =A{e}^{\frac{t}{k}}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}A={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill N& =B{e}^{\frac{t}{k}}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}B\phantom{\rule{1em}{0ex}}is\phantom{\rule{1em}{0ex}}a\phantom{\rule{1em}{0ex}}constant.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Find the family of curves which satisfy the differential equation $\frac{dy}{dx}=-\frac{x}{y}\phantom{\rule{0.3em}{0ex}}.$
 a) ${x}^{2}+{y}^{2}={r}^{2}$ b) $y=\sqrt{{x}^{2}+C}$ c) $y=-\frac{1}{2}{x}^{2}+C$ d) This differential equation cannot be solved using the techniques learned in this course.

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Find the family of curves which satisfy
$\frac{dy}{dx}=3{x}^{2}\left(y-1\right).$
 a) $y=B{e}^{{x}^{3}}-1$ where $B$ is a constant. b) ${y}^{2}-y-{x}^{3}-C=0$ where $C$ is a constant. c) ${y}^{2}-y+{x}^{3}-C=0$ where $C$ is a constant. d) $y=B{e}^{{x}^{3}}+1$ where $B$ is a constant.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\begin{array}{llll}\hfill \frac{dy}{dx}& =3{x}^{2}\left(y-1\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dy}{y-1}& =\int 3{x}^{2}\phantom{\rule{0.3em}{0ex}}dx,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|y-1|& ={x}^{3}+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |y-1|& =A{e}^{{x}^{3}}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}A={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =B{e}^{{x}^{3}}+1\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}B\phantom{\rule{1em}{0ex}}is\phantom{\rule{1em}{0ex}}a\phantom{\rule{1em}{0ex}}constant.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Using Newton’s law of cooling, it is found that the differential equation  describing the heating of a potato in an oven is
$\frac{dT}{dt}=-k\left(T-200\right)\phantom{\rule{0.3em}{0ex}}.$
What is the temperature inside the oven ?
 a) 200k${}^{o}$C b) 200${}^{o}$C c) 21.1${}^{o}$C d) The temperature cannot be determined from this equation.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Use Newton’s law of cooling to determine the temperature with respect to time of a potato cooked in an oven at 200${}^{o}$C, where the temperature of the potato is 20${}^{o}$C shortly before being placed in the oven.
 a) $T=20{e}^{-kt}-200$ b) $T=180{e}^{-kt}-200$ c) $T=21-{e}^{-kt}$ d) $200-180{e}^{-kt}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$\begin{array}{llll}\hfill \frac{dT}{dt}& =-k\left(T-200\right),\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int \frac{dT}{T-200}& =\int -k\phantom{\rule{0.3em}{0ex}}dt,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ln|T-200|& =-kt+C,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill |T-200|& =A{e}^{-kt}\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}A={e}^{C},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill T& =B{e}^{-kt}+200\phantom{\rule{1em}{0ex}}where\phantom{\rule{1em}{0ex}}B\phantom{\rule{1em}{0ex}}is\phantom{\rule{1em}{0ex}}a\phantom{\rule{1em}{0ex}}constant.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$ At $t=0$, $T=20\phantom{\rule{0.3em}{0ex}},$ therefore $20=B+200$ and $B=-180\phantom{\rule{0.3em}{0ex}},$ hence $T=200-180{e}^{-kt}\phantom{\rule{0.3em}{0ex}}.$