Free Printable D.E. Prelim Summative Test

1.

Consider an ODE of the form $G\left(x,f\right)dx\ +H\left(x,f\right)df=0$ , which is not exact. Which function makes the ODE exact when multiplied to the equation?

a)

Common Factor

b)

Integrating Factor

c)

Quadratic Factor

d)

Natural Factor

2.

Consider the differential equation $\frac{\text{d}f}{\text{d}x}+fP\left(x\right)=Q\left(x\right)$ . Let $u=\int_{ }^{ }P\left(x\right)dx$ What is the general solution to the differential equation?

a)

$f=e^{-u}\int_{ }^{ }Q\left(x\right)e^udx$

b)

$f=P\left(u\right)\int_{ }^{ }e^{Q\left(x\right)}dx$

c)

$f=ue^u$

d)

$f=Q\left(u\right)+C$

3.

What do you call the highest derivative occurring in a differential equation?

a)

Degree

b)

Exponent

c)

Order

d)

Power

4.

Let r be a constant. If the equation $\frac{\text{d}f}{\text{d}t}=rf$ represents the model of population growth formulated by Thomas Robert Malthus, what does f(t) represent?

a)

It is the number of new-born babies at any time .

b)

It is a function that gives the number of individuals in a population at any time .

c)

It is the number of deaths occurring at any time .

d)

It is the number of individuals with ages beyond .

5.

What are equations containing a function and its derivatives?

a)

Differential Equation

b)

Functional Equation

c)

Integral Equation

d)

Linear Equation

6.

Which of the following is a differential equation?

a)

$2x\ =\ \frac{\text{d}f}{\text{d}x}$

b)

$\sin\left(2x+1\right)=x^2$

c)

$e^{\left(x-1\right)}=\frac{e^x}{e}$

d)

$\int_{ }^{ }f\left(x\right)dx=3x$

7.

What form can an ODE take if it is separable?

a)

$f\left(G\right)dx=G\left(H\right)df$

b)

$G\left(f\right)=H\left(x\right)+C$

c)

$\frac{\text{d}G}{\text{d}x}=\frac{\text{d}H}{\text{d}x}$

d)

$G\left(f\right)df=H\left(x\right)dx$

8.

Which condition must the differential equation satisfy for it to be exact?

a)

$\frac{\partial H}{\partial G}=\frac{\partial x}{\partial f}$

b)

$\frac{\partial G}{\partial H}=\frac{\partial f}{\partial x}$

c)

$\frac{\partial G}{\partial x}=\frac{\partial H}{\partial f}$

d)

$\frac{\partial G}{\partial f}=\frac{\partial H}{\partial x}$

9.

What method is used in solving homogeneous ODEs?

a)

Transformation of Variables

b)

Separation of Variables

c)

Test of Exactness

d)

Method of Substitution

10.

The difference between the rate at which fluid flows into the substance and the rate at which it flows out is equal to which of the following?

a)

Net rate of change in viscosity of substance at any given time

b)

Net rate of change in amount of substance at any given time

c)

Net rate of change in speed of substance at any given time

d)

Net rate of change in temperature of substance at any given time

11.

Let f be the temperature of the body at any time t, c be the temperature of the environment, and $\Upsilon$ be a constant less than zero. What is the equation for Newton's law of cooling?

a)

$\frac{\text{d}f}{\text{d}t}=f\left(\Upsilon c\right)$

b)

$\frac{\text{d}f}{\text{d}t}=\Upsilon\left(f-c\right)$

c)

$\frac{\text{d}f}{\text{d}c}=f\left(\Upsilon t\right)$

d)

$\frac{\text{d}f}{\text{d}c}=t\left(f-\Upsilon\right)$

12.

Let k be the maximum population an environment can support. What is the mathematical model for the population growth formulated by Pierre Francois Verhulst?

a)

$\frac{\text{d}f}{\text{d}t}=rf\left(1-\frac{f}{k}\right)$ for some constant r

b)

$\frac{\text{d}f}{\text{d}t}=rfk$

for some constant r

c)

$\frac{\text{d}f}{\text{d}t}=\Upsilon\left(f-c\right)$

for some $\Upsilon<0$

d)

$\frac{\text{d}f}{\text{d}t}=k\left(1+f\right)$

13.

What are equations that can be written in the form $\frac{\text{d}f}{\text{d}x}+fP\left(x\right)=Q\left(x\right)$ ?

a)

Linear First-order ODE

b)

Separable ODEs

c)

Homogeneous ODEs

d)

Nonhomogeneous ODEs

14.

What do you call the equations that contain a function in one (1) variable and its derivatives?

a)

Partial Differential Equations (PDEs)

b)

Ordinary Differential Equations (ODEs)

c)

Integral Equations

d)

Mathematical Model

15.

Which principle states that F=ma, where F is net force, m is mass, and a is acceleration?

a)

Aristotelian Theory of Motion

b)

Parallelogram Principle of Force

c)

Law of Inertia

d)

Newton's Second Law of Motion

16.

The concentration of outflowing sand inside a tube is $1.5\ \frac{g}{m^3}$ . If the fluid-out rate is $7\ \frac{m^3}{\min}$ , what is the rate at which the sand flows out of the tube?

a)

8.5 g/min

b)

5.5 g/min

c)

10.5 g/min

d)

4.666... g/min

17.

The concentration of inflowing sand inside a tube is $2\ \frac{g}{m^3}$ . If the fluid-in rate is $7\ \frac{m^3}{\min}$ , what is the rate at which the sand flows into the tube?

a)

9 g/min

b)

5 g/min

c)

14 g/min

d)

3.5 g/min

18.

In a linear first-order ODE of the form $\frac{\text{d}f}{\text{d}x}+fP\left(x\right)=Q\left(x\right)$ , we let $u=\int_{ }^{ }P\left(x\right)dx$ . Consider the linear first-order ODE $\frac{\text{d}f}{\text{d}x}+2fx=2x$ . What is the function $u$ in this ODE?

a)

$u\left(f\right)=2f$

b)

$u\left(f\right)=f^2$

c)

$u\left(x\right)=2x$

d)

$u\left(x\right)=x^2$

19.

Which function is an integrating factor for the ODE $fdx+\left(2x^2f-x\right)df=0$

a)

$M\left(x,f\right)=x^2$

b)

$M\left(x,f\right)=f$

c)

$M\left(x,f\right)=x^{-2}$

d)

$M\left(x,f\right)=f^{-1}$

20.

Consider the differential equation $df=\left(\cos x-f\sin x\right)dx$ . What will be its new form if we transform it into the form $\frac{\text{d}f}{\text{d}x}+fP\left(x\right)=Q\left(x\right)$ ?

a)

$\frac{\text{d}f}{\text{d}x}+f=\cot x$

b)

$\frac{\text{d}f}{\text{d}x}+f=\tan x$

c)

$\frac{\text{d}f}{\text{d}x}+f\left(-\sin x\right)=-\cos x$

d)

$\frac{\text{d}f}{\text{d}x}+f\sin x=\cos x$

21.

Which function is a particular solution to $\frac{\text{d}f}{\text{d}x}=\sin x$ ?

a)

$f\left(x\right)=-\cos x$

b)

$f\left(x\right)=\sin x$

c)

$f\left(x\right)=\tan x$

d)

$f\left(x\right)=\sec x\tan x$

22.

Consider the differential equation $\frac{\text{d}f}{\text{d}x}=\frac{x-f}{x+f}$ What will be its new form if we transform it into the form $\frac{\text{d}f}{\text{d}x}=G\left(\frac{f}{x}\right)$ ?

a)

$\left(x+f\right)\frac{\text{d}f}{\text{d}x}=x-f$

b)

$\left(x+f\right)df=\left(x-f\right)dx$

c)

$\frac{\text{d}f}{\text{d}x}=\frac{1-\frac{f}{x}}{1+\frac{f}{x}}$

d)

$fdf=xdx$

23.

Given a separable differential equation $3\frac{\text{d}f}{\text{d}x}=2x$ what will be its new form if the variables are separated?

a)

$\frac{3}{2x}\cdot\frac{\text{d}f}{\text{d}x}=1$

b)

$\frac{3}{2}\cdot\frac{\text{d}f}{\text{d}x}=x$

c)

$\frac{3}{2}dx=xdf$

d)

$3df=2xdx$

24.

A population in a neighborhood grows by 10% per year. If f(t) is the population at any year t, which differential equation satisfies to the theory of Thomas Robert Malthus?

a)

$\frac{\text{d}f}{\text{d}t}=\frac{1}{100}f\left(t\right)$

b)

$\frac{\text{d}f}{\text{d}t}=\frac{1}{10}f\left(t\right)$

c)

$\frac{\text{d}f}{\text{d}t}=10f\left(t\right)$

d)

$\frac{\text{d}f}{\text{d}t}=100f\left(t\right)$

25.

A force acted on a 10kg object causing it to accelerate by $3\ \frac{m}{s^2}$ . How much is the force that acted on the object?

a)

13 N

b)

7 N

c)

3.333... N

d)

30 N

26.

Let v be the object's velocity at a time t. If a is acceleration, then a is equal to which of the following?

a)

$\frac{\text{d}v}{\text{d}t}$

b)

$\frac{\text{d}^2v}{\text{d}t^2}$

c)

$\frac{\partial v}{\partial a}$

d)

$\frac{\partial^2v}{\partial a\partial t}$

27.

Which of the following is NOT a linear first-order ODE?

a)

$df=\left(x^3fx-f\left(x+1\right)\right)dx$

b)

$df=\left(2x-f\cos x\right)dx$

c)

$df=\left(2\left|x\right|-\frac{f}{e^x}\right)dx$

d)

$df=\frac{2x}{fe^x}dx$

28.

Let $f\left(x\right)=x^2$ . Which differential equation has f as a particular solution?

a)

$\frac{\text{d}f}{\text{d}x}=2x$

b)

$\frac{\text{d}^2f}{\text{d}x^2}=2x$

c)

$\frac{\text{d}^2f}{\text{d}x^2}+\frac{\text{d}f}{\text{d}x}=x^2$

d)

$\frac{\text{d}^3f}{\text{d}x^3}=\frac{1}{3}x^3$

29.

What is the order of $4\frac{\text{d}^2f}{\text{d}x^2}+\frac{\text{d}f}{\text{d}x}=f$ ?

a)

1

b)

4

c)

3

d)

2

30.

Which of the following ODEs is exact?

a)

$2fdx+x^2df=0$

b)

$\left(f+x\right)dx+\left(f-x\right)df=0$

c)

$fdx+\left(x+1\right)df=0$

d)

$fxdx+fx^2df=0$

31.

Consider the exact ODE $\left(2x+f\right)dx+\left(x+2f\right)df=0$ . Find U(x,f) that satisfies $2x+f=\frac{\partial U}{\partial x}$ , and $x+2f=\frac{\partial U}{\partial f}$ .

4 lines

32.

A warm body is submerged in a water with a temperature of $15\degree C$ . Let f(t) be the temperature of the body after minutes of submersion. Find the general solution to Newton's law of cooling.

4 lines

33.

Find the general solution to the linear first-order ODE $\frac{\text{d}f}{\text{d}x}+2fx=2x$ .

4 lines

34.

Give the general solution to the differential equation $\frac{\text{d}f}{\text{d}x}=4-\sin x$ .

4 lines

35.

Find the general solution to $\frac{\text{d}f}{\text{d}x}=3x$ using separation of variables.

4 lines

D.E. Prelim Summative Test

Total questions: 35

Worksheet time: 30mins

Similar Resources on Wayground