## MATH1003 Quizzes

Quiz 12: Systems of Differential Equations
Question 1 Questions
Which of the following functions $x\left(t\right)$ and $y\left(t\right)$ satisfy the differential equations
$\frac{dx}{dt}=2xt;\phantom{\rule{2em}{0ex}}y\frac{dy}{dt}=xt,$ with $x\left(0\right)=4$ and $y\left(0\right)=1$? Exactly one option must be correct)
 a) $x={e}^{{t}^{2}}+3;\phantom{\rule{2em}{0ex}}y=\sqrt{{e}^{{t}^{2}}+3{t}^{2}}$ b) $x=4{e}^{{t}^{2}};\phantom{\rule{2em}{0ex}}y=\sqrt{4{e}^{{t}^{2}}-3}$ c) $x=4{e}^{{t}^{2}};\phantom{\rule{2em}{0ex}}y=2\sqrt{{e}^{{t}^{2}}}-1$ d) $x=4{e}^{{t}^{2}};\phantom{\rule{2em}{0ex}}y=2{e}^{t}-1$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
If $x\left(t\right)$ and $y\left(t\right)$ satisfy the differential equations ${x}^{\prime }=-0.05y;\phantom{\rule{2em}{0ex}}{y}^{\prime }=-0.01x,$
and ${x}^{″}+ax=0$ is the second order differential equation satisfied by $x\left(t\right)$, what is the value of $a$?

Correct!
Note that ${x}^{″}=-0.05{y}^{\prime }$.
If $x\left(t\right)$ and $y\left(t\right)$ satisfy the differential equations $\frac{dx}{dt}=-2x+3y;\phantom{\rule{2em}{0ex}}\frac{dy}{dt}=7x-5y,$
which of the following is the second order differential equation satisfied by $x\left(t\right)$? Exactly one option must be correct)
 a) $\frac{{d}^{2}x}{d{t}^{2}}+7\frac{dx}{dt}-11x=0$ b) $\frac{{d}^{2}x}{d{t}^{2}}-9\frac{dx}{dt}-2x=0$ c) $\frac{{d}^{2}x}{d{t}^{2}}+7\frac{dx}{dt}+5x=0$ d) $\frac{{d}^{2}x}{d{t}^{2}}-7\frac{dx}{dt}-4x=0$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
If $x\left(t\right)$ and $y\left(t\right)$ satisfy the differential equations ${x}^{\prime }=2x-3y;\phantom{\rule{2em}{0ex}}{y}^{\prime }=-x+y,$
which of the following is the second order differential equation satisfied by $y\left(t\right)$? Exactly one option must be correct)
 a) ${y}^{″}-3{y}^{\prime }+y=0$ b) ${y}^{″}+3{y}^{\prime }+y=0$ c) ${y}^{″}-3{y}^{\prime }-y=0$ d) None of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choose the option which is a general solution to the system of differential equations:
$\frac{dx}{dt}=4x-3y;\phantom{\rule{2em}{0ex}}\frac{dy}{dt}=2x-y.$
In each case, $A$ and $B$ are arbitrary constants. Exactly one option must be correct)
 a) $x=A{e}^{t}+B{e}^{2t};\phantom{\rule{1em}{0ex}}y=5A{e}^{t}+6B{e}^{2t}$ b) $x=A{e}^{t}+B{e}^{2t};\phantom{\rule{1em}{0ex}}y=A{e}^{t}+B{e}^{2t}$ c) $x=A{e}^{t}+B{e}^{2t};\phantom{\rule{1em}{0ex}}y=3A{e}^{t}+2B{e}^{2t}$ d) $x=A{e}^{t}+B{e}^{2t};\phantom{\rule{1em}{0ex}}y=A{e}^{t}+\frac{2B}{3}{e}^{2t}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choose the option which is a general solution to the system of differential equations:
$\frac{dx}{dt}=-2x+y;\phantom{\rule{2em}{0ex}}\frac{dy}{dt}=-x-4y.$
In each case, $A$ and $B$ are arbitrary constants. Exactly one option must be correct)
 a) $x={e}^{-3t}\left(At+B\right);\phantom{\rule{1em}{0ex}}y={e}^{-3t}\left(A-At-B\right)$ b) $x={e}^{-3t}\left(At+B\right);\phantom{\rule{1em}{0ex}}y={e}^{-3t}\left(At+B\right)$ c) $x=A{e}^{t}+B{e}^{7t};\phantom{\rule{1em}{0ex}}y=A{e}^{t}+B{e}^{7t}$ d) $x=A{e}^{t}+B{e}^{7t};\phantom{\rule{1em}{0ex}}y=3A{e}^{t}+9B{e}^{7t}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choose the options which are particular solutions to the system of differential equations:
$\frac{dx}{dt}=x+5y;\phantom{\rule{2em}{0ex}}\frac{dy}{dt}=-x-y.$
More than one option my be correct. (Zero or more options can be correct)
 a) $x=cos2t;\phantom{\rule{1em}{0ex}}y=-\frac{2}{5}sin2t-\frac{1}{5}cos2t$ b) $x=cos2t;\phantom{\rule{1em}{0ex}}y=sin2t+cos2t$ c) $x=sin2t;\phantom{\rule{1em}{0ex}}y=-\frac{1}{5}sin2t+\frac{2}{5}cos2t$ d) $x=5cos2t;\phantom{\rule{1em}{0ex}}y=-2sin2t-cos2t$

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be True.
Correct!
1. True
2. False
3. True
4. True
In a certain environment live 10000 particularly harmful insects of species $X$. In the hope of eradicating species $X$, 1000 insects of species $Y$, which eat species $X$, are introduced. Let $X\left(t\right)$ be the number of thousands of insects of species $X$, and $Y\left(t\right)$ be the number of thousands of insects of species $Y$, at time $t$ (in years) after species $Y$ is introduced. Suppose the sizes of the populations are governed by
${X}^{\prime }=5X-Y;\phantom{\rule{2em}{0ex}}{Y}^{\prime }=5X+Y.$
According to this model, which of the following options is correct? Exactly one option must be correct)
 a) Both species survive indefinitely. b) Both species survive for approximately 2.7 years, at which time species $X$ is eradicated. c) Both species survive for approximately 3.2 years, at which time species $Y$ dies out. d) Both species survive for approximately 153 years, at which time species $X$ is eradicated.

Choice (a) is incorrect
The population sizes are given by $X={e}^{3t}\left(10cost+19sint\right),\phantom{\rule{1em}{0ex}}Y={e}^{3t}\left(cost+48sint\right)$.
Look for the smallest value of $t$ that makes $X$ or $Y$ equal to zero.
Choice (b) is correct!
The population sizes are given by $X={e}^{3t}\left(10cost+19sint\right),\phantom{\rule{1em}{0ex}}Y={e}^{3t}\left(cost+48sint\right)$.
Choice (c) is incorrect
The population sizes are given by $X={e}^{3t}\left(10cost+19sint\right),\phantom{\rule{1em}{0ex}}Y={e}^{3t}\left(cost+48sint\right)$.
$X$ is equal to zero before $t=3.2$.
Choice (d) is incorrect
The population sizes are given by $X={e}^{3t}\left(10cost+19sint\right),\phantom{\rule{1em}{0ex}}Y={e}^{3t}\left(cost+48sint\right)$.
Look for the smallest value of $t$ in RADIANS that makes $X$ or $Y$ equal to zero.
Questions 9 and 10 use the following model for a battle between two armies, $X$ and $Y$. The model assumes that the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the number of soldiers in the opposing army. For one particular battle, the proposed model is $\frac{dX}{dt}=-0.04Y;\phantom{\rule{2em}{0ex}}\frac{dY}{dt}=-0.01X$
where $X\left(t\right)$ and $Y\left(t\right)$ are the numbers of soldiers still fighting in armies $X$ and $Y$, respectively, $t$ days after the battle starts. The initial strength of army $X$ was 54 000, and that of army $Y$ was 21 500. If army $Y$ fought without surrendering until all its soldiers were killed how long did the battle last? (Give your answer rounded up to the nearest day.)

Correct!
The particular solution for $Y$ is $Y=24250{e}^{-0.02t}-2750{e}^{0.02t}$. Solving $Y=0$ gives $t=55$.
Questions 9 and 10 use the following model for a battle between two armies, $X$ and $Y$. The model assumes that the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the number of soldiers in the opposing army. For one particular battle, the proposed model is $\frac{dX}{dt}=-0.04Y;\phantom{\rule{2em}{0ex}}\frac{dY}{dt}=-0.01X$
where $X\left(t\right)$ and $Y\left(t\right)$ are the numbers of soldiers still fighting in armies $X$ and $Y$, respectively, $t$ days after the battle starts. The initial strength of army $X$ was 54 000, and that of army $Y$ was 21 500.
Use the answer to question 9 to determine the approximate number of soldiers left in army $X$ when the battle is over. (Give your answer to the nearest integer.)
The particular solution for $X$ is $X=5500{e}^{0.02t}+48500{e}^{-0.02t}$. Substitute $t=55$.