Free Practice Quiz Question List

Differential Equations Online Quiz Questions

Use this free practice quiz with 20 questions to review Differential Equations, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

A differential equation directly specifies a relationship involving an unknown function and one or more of its derivatives.

A
True
B
False

Choose one

1 point

A population’s rate of increase is proportional to its current size. Which differential equation models this statement?

A
$dP/dt=k+P$
B
$dP/dt=kP$
C
$dP/dt=k/P$
D
$dP/dt=P^k$

Fill in the blank

1 point

For the differential equation $dy/dx=x-y$ , what slope is assigned to the point $(1,2)$ ? Enter the numerical slope as .

Written response

1 point

For $dy/dx=x-y$ , enter the slope at the point $(1,2)$ .

Choose one

1 point

An autonomous population equation has the form $dP/dt=f(P)$ . What feature should its slope field have?

A
The slope pattern repeats vertically because the slope depends only on time.
B
The slope changes only along lines where $x+y$ is constant.
C
The slope pattern repeats horizontally because the slope depends only on $P$ .
D
Every solution curve in the field must be a horizontal line.

Fill in the blank

1 point

The separated equation $dy/dx=2xy$ has the general solution $y=Ce^{x^2}$ . If $y(0)=3$ , the integration constant is .

Written response

1 point

Use one Euler step with $h=0.1$ to approximate $y(0.1)$ for $dy/dx=x+y$ with $y(0)=1$ .

Choose one

1 point

A population has carrying capacity $K=1000$ , initial population $P(0)=100$ , and logistic growth rate $r=0.4$ per year. Which function models the population?

A
$P(t)=1000/(1+0.1e^{-0.4t})$
B
$P(t)=1000/(1+9e^{-0.4t})$
C
$P(t)=1000/(1+9e^{0.4t})$
D
$P(t)=100/(1+9e^{-0.4t})$

Choose all

1 point

For the logistic equation $dP/dt=rP(1-P/K)$ , where $r>0$ and $K>0$ , select all statements that are true.

A
$P=0$ and $P=K$ are equilibrium solutions.
B
If $0<P<K$ , then the population increases.
C
If $P>K$ , then the population decreases.
D
The growth rate is largest when $P=K/2$ .

True or false

1 point

For a fixed interval, decreasing the step size in Euler’s method generally improves the approximation but requires more calculations.

A
True
B
False

Choose all

1 point

Select all statements that correctly compare exponential and logistic population models.

A
An exponential model assumes unlimited resources.
B
A logistic population grows without bound whenever its rate parameter is positive.
C
A logistic model approaches a carrying capacity in the long term.
D
Logistic growth is fastest at half the carrying capacity.

Open ended

1 point

Explain a general procedure for building and interpreting a differential-equation model. Include how the statement “a population grows at a rate proportional to its current size” is translated and what the sign of the proportionality constant means.

Choose one

1 point

A quantity follows the exponential model $P(t)=P_0e^{0.2t}$ , where $t$ is measured in hours. What is its doubling time?

A
Approximately 0.14 hours
B
Approximately 1.39 hours
C
Approximately 3.47 hours
D
Approximately 6.93 hours

Choose one

1 point

A population P(t) increases at a rate proportional to its current size. Which differential equation models this statement?

A
dP/dt = k + P
B
dP/dt = kP
C
dP/dt = k/P
D
dP/dt = P/k

True or false

1 point

True or false: An initial condition can identify one particular solution from a family of solutions to a differential equation.

A
True
B
False

Choose one

1 point

What characteristic should a slope field have for an autonomous equation of the form dP/dt=f(P)?

A
The slope pattern repeats vertically because the slope depends only on time.
B
The slope pattern is random because autonomous equations have no fixed slopes.
C
The slope pattern repeats horizontally because the slope depends only on P.
D
Every solution curve must be a straight line because the equation is autonomous.

Written response

1 point

Use Euler’s method with step size h=0.1 to approximate y(0.3) for dy/dx=x+y with y(0)=1. Enter the numerical approximation to three decimal places.

Written response

1 point

In a logistic population model, what term describes the maximum sustainable population represented by K?

Choose one

1 point

For the logistic equation dP/dt = rP(1−P/K), at which population value is the growth rate greatest?

A
P=0
B
P=K/4
C
P=K
D
P=K/2

Choose one

1 point

Which recurrence correctly describes Euler’s method for dy/dx=f(x,y) with step size h?

A
x_(n+1)=x_n+h and y_(n+1)=y_n+h f(x_n,y_n)
B
x_(n+1)=x_n+h f(x_n,y_n) and y_(n+1)=y_n+h
C
x_(n+1)=x_n−h and y_(n+1)=y_n−h f(x_n,y_n)
D
x_(n+1)=x_n+h and y_(n+1)=y_n+f(x_n,y_n)/h