## MATH1013 Quizzes

Quiz 8: First order differential equations and slope fields
Question 1 Questions
Which of the following differential equations are first order differential equations ?
 a) $\frac{{d}^{2}y}{d{t}^{2}}+t\frac{dy}{dt}={t}^{2}y=sint$ b) ${x}^{2}\frac{dx}{dt}+x=sint$ c) $\frac{dy}{dx}=4{x}^{2}$ d) $\frac{{d}^{2}y}{d{x}^{2}}=4y$

There is at least one mistake.
For example, choice (a) should be False.
This is a second order differential equation.
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 True.
There is at least one mistake.
For example, choice (d) should be False.
This is a second order differential equation.
Correct!
1. False This is a second order differential equation.
2. True
3. True
4. False This is a second order differential equation.
Consider the first order differential equation $\frac{dy}{dt}=4y$ where $y=1000$ at $t=0\phantom{\rule{0.3em}{0ex}}.$ What is the general solution to the differential equation ?
 a) $y=1000{e}^{4t}$ b) $y={t}^{4}+C$ c) $y=B{e}^{4t}$ d) $y={t}^{4}+1000$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Note that the first response is the particular solution to the differential equation with the given initial condition.
Choice (d) is incorrect
Suppose you know that $Q=C{e}^{kt}$ satisfies the differential equation $\frac{dQ}{dt}=-2Q\phantom{\rule{0.3em}{0ex}}.$ What does this tell you about the values of $C$ and $k$?
 a) $C=-2$ and $k=1$. b) $C=-1$ and $k=2$. c) $C=-2$ and we know nothing of the value of $k$. d) $k=-2$ and we know nothing of the value of $C$.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$Q=C{e}^{kt}\phantom{\rule{1em}{0ex}}⇒\frac{dQ}{dt}=Ckekt=kC{e}^{kt}=kQ\phantom{\rule{0.3em}{0ex}}.$
Hence $k=-2$ but we know nothing of the value of $C$.
For what values of $\lambda$ is $y=A{x}^{\lambda }$ a solution of the differential equation
${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+2x\frac{dy}{dx}-6y=0\phantom{\rule{0.3em}{0ex}}?$
 a) $\lambda =2$ and $\lambda =-3$. b) $\lambda =-2$ and $\lambda =3$. c) $\lambda =-1+\sqrt{7}$ and $\lambda =-1-\sqrt{7}$. d) There is not enough information to tell.

Choice (a) is correct!
If $y=A{x}^{\lambda }$ then $\frac{dy}{dx}=\lambda A{x}^{\lambda -1}$ and $\frac{{d}^{2}y}{d{x}^{2}}=\lambda \left(\lambda -1\right)A{x}^{\lambda -2}\phantom{\rule{0.3em}{0ex}}.$ Hence $\begin{array}{llll}\hfill & {x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+2x\frac{dy}{dx}-6y=0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ⇒\phantom{\rule{1em}{0ex}}& {x}^{2}\lambda \left(\lambda -1\right)A{x}^{\lambda -2}+2x\lambda A{x}^{\lambda -1}-6A{x}^{\lambda }=0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ⇒\phantom{\rule{1em}{0ex}}& \lambda \left(\lambda -1\right)A{x}^{\lambda }+2\lambda A{x}^{\lambda }-6A{x}^{\lambda }=0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ⇒\phantom{\rule{1em}{0ex}}& \lambda \left(\lambda -1\right)+2\lambda -6=0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ⇒\phantom{\rule{1em}{0ex}}& {\lambda }^{2}+\lambda -6=\left(\lambda +3\right)\left(\lambda -2\right)=0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill ⇒\phantom{\rule{1em}{0ex}}& \lambda =-3\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}\lambda =2\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the following direction (slope) field.
Which differential equation corresponds to this direction field ?
 a) $\frac{dy}{dx}=1+{y}^{2}$ b) $\frac{dy}{dx}=x$ c) $\frac{dy}{dx}=x-y$ d) $\frac{dy}{dx}=4-y$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Consider the following direction (slope) field.
Which differential equation corresponds to this direction field ?
 a) $\frac{dy}{dx}=sinx$ b) $\frac{dy}{dx}=-y$ c) $\frac{dy}{dx}=-cosx$ d) $\frac{dy}{dx}=x$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which of the direction fields below correspond to the differential equation $\frac{dy}{dt}=3-y?$
 a) b) c) d)

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Which if the direction fields below correspond to the differential equation $\frac{dy}{dt}=-t$?
 a) b) c) d)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Consider the direction field below for the differential equation ${y}^{\prime }=1+{y}^{2}$ with the four curves labeled I to IV.
Which is the graph of the solution curve passing through (2,4) ?
 a) Curve I b) Curve II c) Curve III d) Curve IV

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Consider the direction field for ${y}^{\prime }=\frac{1}{2}\left(1+y\right)\left(2-y\right)$ below.
Which points could possibly lie on the same solution curve ?
 a) $\left(-1,-1\right)\phantom{\rule{0.3em}{0ex}}$, $\left(0,0\right)\phantom{\rule{0.3em}{0ex}}$, $\left(1,1\right)\phantom{\rule{0.3em}{0ex}}$, $\left(1,2\right)\phantom{\rule{0.3em}{0ex}}.$ b) $\left(-1,-1\right)\phantom{\rule{0.3em}{0ex}}$, $\left(0,\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}$, $\left(1,\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}$, $\left(2,5\right)\phantom{\rule{0.3em}{0ex}}.$ c) $\left(-1,-4\right)\phantom{\rule{0.3em}{0ex}}$, $\left(0,-3\right)\phantom{\rule{0.3em}{0ex}}$, $\left(1,-2\right)\phantom{\rule{0.3em}{0ex}}$, $\left(2,1\right)\phantom{\rule{0.3em}{0ex}}.$ d) $\left(-1,-\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}$, $\left(0,\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}$, $\left(1,\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}$, $\left(2,1\right)\phantom{\rule{0.3em}{0ex}}.$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!