Free Printable Differential Equations Exam

1.

Given the differential equation dP/Dt=5P

A) Find the general solution for P to the differential equation

B)Find the particular solution for P to the differential equation given P(0)= 418

a)

A) P=Ce^5t

B) P= 5e^418t

b)

A) P=Ce^5t

B) P= 418e^5t

c)

A) P=Ce^t

B) P= 418e^t

d)

A) P=Ce^10t

B) P= 418e^10t

2.

The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year.

Find the general solution.

a)

P=.05e^1000t

b)

P=1000e^5t

c)

P=1000e^.05t

d)

p=lne^1000t

3.

Y is proportional to the difference of x and z

a)

y= x-z

b)

y= k(z-x)

c)

y= k(x-z)

d)

y= z-x

4.

dy/dx = 2x/e^2y find the general solution

a)

y = ln (2x/2 + C)

2

b)

y = ln(2x/2)+c

c)

y = e^2x+c

d)

y = mx+b

5.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

𝑦 = ln(15t)

b)

𝑦 = 16t³

c)

𝑦 = (2𝑡³− 16)^1/2

d)

𝑦 = (2𝑡³−16)

6.

dy/dx= x √y

Find the general solution

a)

√y = ln|4x| + c

b)

√y = (x)^-1 + c

c)

2y^1/2= ln|x| + c

d)

1/2y^1/2= ln|x| + c

7.

y varies jointly with x and the square of z

a)

y= kxz²

b)

y= kx√z

c)

y= kxz

d)

kx²z

8.

what would the sign of the square root after finding c be given the initial condition g(1)= -5

a)

positive

b)

negative

9.

Dy/dx= tanx+15x2+e^x+1/x

a)

y= -ln|sinx| + 5x³ + e^x + ln|x| + c

b)

y= secxtanx + 5x³ + e^x + ln|x| + c

c)

y= -ln|cosx| + 5x³ + e^x + ln|x| + c

d)

y= sec²(x) + 5x³ + e^x + ln|x| + c

10.

dp/dy = (4y)^-1find the general solution

a)

p= 1/5y^5/4 + c

b)

p= 1/4ln|4y| + c

c)

p= 1/8y² + c

d)

p= 1 + c

4y

11.

y is proportional to the product of z and the square root of x

a)

y= z(x²)

b)

y= kz(x²)

c)

y= kx√z

d)

y=zkx

12.

dy/dx = 4x/y. Suppose y(0)=1

The particular solution is

a)

B

b)

C

c)

D

d)

E

13.

Find the particular solution for y if dy/dx = 2x√y and y = 4 when x = 3.

a)

2√y = x² + C

b)

y = (x² + 25)²/4

c)

y = (x²/2-5/2)^1/2

d)

y = x⁴/4

14.

How would you CORRECTLY separate the following differential equation?

$\frac{dy}{dx}=\frac{2x+1}{3y}$

a)

$3y\cdot dy=\left(2x+1\right)dx$

b)

$\frac{dy}{3y}=\left(2x+1\right)dx$

c)

$\frac{dy}{3y}=\frac{dx}{2x+1}$

d)

$3y^{-1}dy=\left(2x+1\right)dx$

15.

Now that we have separated, what would be the correct integration?

$3y\cdot dy=\left(2x+1\right)dx$

a)

$3\ln\left|y\right|=x^2+x+C$

b)

$\frac{3}{2}y^2dy=x^2+x$

c)

$\frac{3}{2}y^2=x^2+x+C$

d)

$3y^2=\frac{1}{2}x^2+x+C$

16.

How would you CORRECTLY separate the following differential equation?

$\frac{dy}{dx}=2y-xy$

a)

$\frac{1}{y}dy=\left(2-x\right)dx$

b)

$y\cdot dy=\left(2-x\right)dx$

c)

$\frac{dy-2y}{y}=xdx$

d)

$\frac{-2y+dy}{y}=x\cdot dx$

17.

$\frac{1}{y}dy=\left(2-x\right)dx$ Now that we have separated, what would be the correct integration?

a)

$\ln\left(y\right)=x-x^2+C$

b)

$\ln\left|y\right|=2x-\frac{1}{2}x^2+C$

c)

$-\frac{1}{2}y^{-2}=x-\frac{1}{2}x^2+C$

d)

$\ln\left|y\right|=\frac{1}{2}x^2-2x+C$

18.

How would you CORRECTLY separate the following differential equation?

$\frac{dy}{dx}=e^{\left(x+y\right)}$

a)

$e^{-y}dy=e^xdx$

b)

$-\frac{dy}{e^y}=e^xdx$

c)

$\frac{dy}{e^{\left(x+y\right)}}=1dx$

d)

$1dy=e^{\left(x+y\right)}dx$

19.

Now that we have separated, what would be the correct integration?

$e^{-y}dy=e^xdx$

a)

$e^y=e^x+c$

b)

$e^{-y}=e^x+c$

c)

$-e^{-y}=-e^x+c$

d)

$-e^{-y}=e^x+C$

20.

In drawing the slope field for the differential equation , I would place short slope lines of _________ at the points (0,1), (1,-1), and (2,-2)
$\frac{dy}{dx}=2x-3y$

Select all that apply

a)

-3

b)

-1

c)

2

d)

5

e)

10

21.

Which choice below represents this slope field?

a)

dy/dx = 2x

b)

dy/dx = -x

c)

dy/dx = yx

d)

dy/dx = x²

22.

Which equation below represents the slope field?

a)

dy/dx = x - 2

b)

dy/dx = 1/2x + 1

c)

dy/dx =1/2 y - 2

d)

dy/dx = y + 2

23.

Which functions below will have the same slope field? (choose all that apply)

a)

y = x² - 3

b)

y = x² + 4

c)

y = 2x - 5

d)

y = 2x²

24.

Which slope field is represented by dy/dx = x²?

a)

b)

c)

d)

25.

A puppy gains weight, w, at a rate approximately inversely proportional to its age, t, in months.

a)

A

b)

B

c)

C

d)

D

26.

dy/dx = 4x/y. Suppose y(0)=1
The particular solution is

a)

B

b)

C

c)

D

d)

E

27.

Water flows continuously from a large tank at a rate proportional to the amount of water in the tank, modeled by dy/dt = ky. There was initially 10,000 ft^3 at t=0. After 4 hours there were 8000 ft^3 remaining.
What is the value of k in the differential equation (calculator)

a)

-0.050

b)

-0.056

c)

-.169

d)

-.200

28.

Consider the differential equation dy/dx = x + 2y for which g(x) is the solution. Which of the following statements is true if the particular solution contains (0,-1)

a)

g(x) is increasing and concave up

b)

g(x) is increasing and concave down

c)

g(x) is decreasing and concave up

d)

g(x) is decreasing and concave down

29.

Y is proportional to the difference of x and z

a)

y= x-z

b)

y= k(z-x)

c)

y= k(x-z)

d)

y= z-x

30.

y is proportional to the product of z and the square root of x

a)

$y=zx^2$

b)

$y=kzx^2$

c)

$y=kz\sqrt{x}$

d)

$y=zkx$

31.

∫dx

a)

x+c

b)

2x+c

c)

3x+c

d)

4x+c

32.

Find the mistake if possible:

a)

The Diff EQ is solved correctly

b)

Step 1 is incorrect. The separation of variables wasn't done correctly.

c)

Step 2 is incorrect. They didn't integrate x² correctly.

d)

Step 3 is incorrect. They didn't take the reciprocal of x³/3+C correctly.

33.

Find f(1) given f'(x)

a)

-12

b)

0

c)

14

d)

20

34.

Evaluate:

a)

4

b)

6

c)

8

d)

10

35.

Find the Particular Solution

a)

$y=2+e^{\left(\frac{x^2}{2}+x\right)}$

b)

$y=2e^{\left(\frac{x^2}{2}+x\right)}$

c)

$y=\ln\left|\frac{x^2}{2}+x+1\right|+2$

d)

$y=\ln\left|\frac{x^2}{2}+x+e^2\right|$

36.

Solve the Initial Value Problem

a)

$y=-\frac{1}{x}$

b)

$y=-x^2$

c)

$y=\frac{-1}{x+1}$

d)

$y=\frac{1}{x+1}$

37.

Of the following, which is a solution to the differential equation, select all that apply:
$y''-6y'+8y=0$

a)

$y=2\sin\left(4x\right)$

b)

$y=3e^{2x}$

c)

$y=Ce^{4x},\$ where C is a constant

38.

An equation that contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable

a)

Dependent Differential Equation

b)

First Order Differential Equation

c)

Partial Differential Equation

d)

Ordinary Differential Equation

39.

An equation containing derivatives of one or more dependent variables with one or more independent variables?

a)

Partial Equation

b)

Different Equation

c)

Differential Equation

d)

Ordinary Equation

40.

An equation containing partial derivatives of one or more dependent variables of two or more independent variables

a)

Ordinary Partial Differential Equation

b)

Partial Differential Equation

c)

Ordinary Differential Equation

d)

Partially Ordinary Differential Equation

41.

\frac{\text{d}x}{\text{d}y}=\cos\ x

is an ordinary differential equation

a)

True

b)

False

42.

a)

is a differential equation

b)

is not a differential equation

c)

either ordinary or partial differential equation

d)

neither ordinary of partial differential equation

43.

it is number of the highest degree in a differential equation

a)

variable

b)

degree

c)

order

d)

derivative

44.

The power of the differential equation's highest derivative, after the equation has been made rational and integral in all of its derivatives

a)

order

b)

degree

c)

exponent

d)

variable

45.

which is true?

a)

it is a partial differential equation

b)

it is an ordinary differential equation

c)

it is not a differential equation

d)

it is an equation that cannot be differentiated

46.

when the dependent variable and all of its derivatives are of the first degree, the equation is

a)

non linear

b)

linear

c)

both linear and non linear

d)

none of the options

47.

an equation is non-linear if the coefficients depend at most on the "independent" variables

a)

True

b)

False

48.

in $\frac{\text{d}x}{\text{d}y}$ , which is the dependent variable?

a)

x

b)

y

c)

d

d)

secret

49.

in $\frac{\text{d}y}{\text{d}x}$ , the independent variable is

a)

x

b)

y

c)

d

d)

none of the options syempre!

50.

$\frac{\partial x}{\partial y}$ is a partial derivative

a)

Yes

b)

No

c)

neither

d)

Either

51.

Check all ordinary differential equation

a)

$\frac{\text{d}x}{\text{d}t}=5x-3$

b)

$\frac{\partial x}{\partial y}-2=3xy$

c)

$\left(x^2+y^2\right)dx-2xydy=0$

d)

$\left(\frac{\partial^3W}{\partial x^2}\right)^3=xy\frac{\partial^2W}{\partial y^2}+\frac{\partial^2W}{\partial z^2}=0$

52.

Check all partial differential equation

a)

$\frac{\text{d}x}{\text{d}t}=5x-3$

b)

$\frac{\partial x}{\partial y}-2=3xy$

c)

$\left(x^2+y^2\right)dx-2xydy=0$

d)

$\left(\frac{\partial^3W}{\partial x^2}\right)^3=xy\frac{\partial^2W}{\partial y^2}+\frac{\partial^2W}{\partial z^2}=0$

53.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

$y=\ln\left|15t\right|$

b)

$y=16t^3$

c)

$y\ =\ \sqrt{2t^3-16}$

d)

$y=2t^3-16$

54.

Determine the value of "c" that satisfies the differential equation $\frac{dy}{dx}=\frac{x+1}{y+2}$ if the curve goes through the point (0, -1).

a)

5/2

b)

-3/2

c)

-1/2

d)

1

55.

Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^2}{y}$ with the initial condition y(3) = -2?

a)

$y=-2e^{\left(-9+\frac{x^3}{3}\right)}$

b)

$y=\sqrt{\frac{2x^3}{3}}$

c)

$y=\sqrt{\frac{2x^3}{3}-14}$

d)

$y=-\sqrt{\frac{2x^3}{3}-14}$

56.

Given the differential equation dP/Dt=5P

A) Find the general solution for P to the differential equation

B)Find the particular solution for P to the differential equation given P(0)= 418

a)

A) P=Ce^5t

B) P= 5e^418t

b)

A) P=Ce^5t

B) P= 418e^5t

c)

A) P=Ce^t

B) P= 418e^t

d)

A) P=Ce^10t

B) P= 418e^10t

57.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$

a)

$\sin y=1+C$

b)

$\sin y=x+C$

c)

$-\sin y=\frac{x^2}{2}+C$

d)

$\sin y=0+C$

58.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=e^x$

a)

$1=e^x+C$

b)

$y=\frac{e^x}{2}+C$

c)

$y=e^x+C$

d)

$0=e^x+C$

59.

Let $\frac{dy}{dx}=-0.4y$ and $y=5\$ when $x=0$ .

What is the solution for $y$ ?

a)

$y=5e^{-0.4x}$

b)

$y=-0.4e^{5x}$

c)

$y=5e^{-\frac{x}{0.4}}$

d)

$y=0.4e^{\frac{x}{5}}$

60.

Solve the differential equation $\frac{dy}{dt}=\frac{3t^2}{y}$ with initial condition 𝑦(2) = 0.

a)

𝑦 = ln(15t)

b)

𝑦 = 16t³

c)

𝑦 = (2𝑡³− 16)^1/2

d)

𝑦 = (2𝑡³−16)

61.

Degree of Differential equation is

a)

Power of highest order derivative

b)

Highest power

c)

Highest order derivative in DE

d)

None of above

62.

I.F. of $\frac{\text{d}x}{\text{d}y}+Px=Q$ is given by:

a)

$e^{\int Pdx_{ }}$

b)

$e^{\int Pdy_{ }}$

c)

$e^{\int Qdy_{ }}$

d)

$e^{\int Qdx_{ }}$

63.

The following Differential Equation is
$\frac{\text{d}y}{\text{d}x}=x^2y+x$

a)

Separable.

b)

Non Separable.

64.

y varies jointly with x and the square of z

a)

y= kxz²

b)

y= kx√z

c)

y= kxz

d)

kx²z

65.

Which choice below represents this slope field?

a)

dy/dx = 2x

b)

dy/dx = -x

c)

dy/dx = x

d)

dy/dx = x²

66.

What is DE

a)

Contains variables

b)

Have constants

c)

Contains variables and its derivatives

d)

Has image

67.

What do mean by linear

a)

Degree 1

b)

Degree 2

c)

Degree 3

d)

Degree 4

68.

What is homogeneous

a)

same degree

b)

same order

c)

different degree

d)

different order

69.

dy/dx = 4x/y. Suppose y(0)=1

The particular solution is

a)

B

b)

C

c)

D

d)

E

70.

An equation containing partial derivatives of one or more dependent variables of two or more independent variables

a)

Ordinary Partial Differential Equation

b)

Partial Differential Equation

c)

Ordinary Differential Equation

d)

Partially Ordinary Differential Equation

71.

Choose the Differential Equations

a)

$3x-4y+xy=1$

b)

$5y+3y''-x^2=1$

c)

$\frac{x}{y}+1=xy$

d)

$6x^3+2y^4=xy-4$

72.

State the order and degree of this following Differential Equations

$\left(\frac{\text{d}^3x}{\text{d}y^3}\right)^5+xy=1-x$

a)

Order=5

Degree=3

b)

Order=1

Degree=1

c)

Order=1

Degree=0

d)

Order=3

Degree=5

73.

Methods to solve DE are.......................

a)

Integration and Differentiation

b)

Integration and Separable Variable Method

c)

Integrating Factor and Separable Variable Methods

d)

Separable Variable Methods only

74.

State the order of given Differential Equations $y^{\left(5\right)}$

a)

5

b)

1

c)

5 and 1

d)

zero

75.

State TWO types of solutions of Differential Equations

a)

General Solutions Only

b)

General and Particular Solutions

c)

Particular Solutions Only

d)

No solutions

76.

Solve the given DE by using Separable Variable Method

$4y^3\ dy\ =\ 2x\ dx$

a)

$4y^4=x^2+c$

b)

$y^4=2x+c$

c)

$y^4=x^2+c$

d)

$\frac{4y^3}{3}=x^2+c$

77.

Solves this following DE by using Separable Variable Method

$y^5\frac{\text{d}y}{\text{d}x}=e^x$

a)

$y^6=e^x+c$

b)

$\frac{y^6}{6}=e^x+c$

c)

$y^5=\frac{e^x}{x}+c$

d)

$\frac{y^6}{5}=xe^x+c$

78.

Given a differential equation as $y\frac{\text{d}y}{\text{d}x}=3$ and y(0)=1. Find the particular solution of this equation by using Separable Variable Method.

a)

$\frac{y^2}{2}=3x+\frac{1}{2}$

b)

$y^2=3+\frac{1}{2}$

c)

$\frac{y^2}{2}=3x+c$

d)

$\frac{y^2}{2}=\frac{3x^2}{3}+\frac{1}{2}$

79.

IVP stand for.............

a)

Index Value Principle

b)

Initial Value Principle

c)

Initial Value Problem

d)

Index Value Problem

80.

Separate this given variable

$\left(xy+4y\right)\frac{dy}{dx}=1$

a)

$4y\ dy\ =\ 1-xy\ dx$

b)

$y\ dy=\frac{\text{d}x}{x+4}$

c)

$y\ dy=\left(x+4\right)dx$

d)

$\left(x+4\right)dy=\frac{dx}{y}$

81.

Which of the following differential equations match the graph above?

a)

$\frac{\text{d}y}{\text{d}x}=y^2-1$

b)

$\frac{\text{d}y}{\text{d}x}=x^2-1$

c)

$\frac{\text{d}y}{\text{d}x}=y^2+1$

d)

$\frac{\text{d}y}{\text{d}x}=x^2+1$

82.

Suppose y=f(x) is a particular solution to the differential equation dy/dx=x–y such that f(0)=0. Use the slope field above to estimate the value of f(2).

a)

-2

b)

-1

c)

0

d)

1

83.

Match the following differential equation with the correct slope field: dy/dx=x–y^2.

a)

b)

c)

d)

84.

Shown above is a slope field for which of the following differential equations?

a)

$\frac{dy}{dx}=1+x$

b)

$\frac{dy}{dx}=x^2$

c)

$\frac{dy}{dx}=x+y$

d)

$\frac{dy}{dx}=\frac{x}{y}$

85.

Which of the following is the solution to the differential equation dy/dx=(x^2)/y with initial condition y(3)=-2?

a)

$y=2e^{\frac{-9+x^3}{3}}$

b)

$y=-2e^{\frac{-9+x^3}{3}}$

c)

$y=\sqrt{\frac{2x^3}{3}}$

d)

$y=\sqrt{\frac{2x^3}{3}-14}$

e)

$y=-\sqrt{\frac{2x^3}{3}-14}$

86.

Solve the differential equation dy/dx=e^2x in terms of y given the condition that y(0)=0.

a)

$y=\frac{1}{2}e^{2x}+\frac{1}{2}$

b)

$y=\frac{1}{2}e^{2x}-\frac{1}{2}$

c)

$y=\frac{1}{2}e^{2x}+\frac{1}{3}$

d)

$y=\frac{1}{2}e^{2x}-\frac{1}{3}$

e)

$y=\frac{1}{2}e^x-\frac{1}{2}$

87.

Solve the differential equation dy/dx=y/1+x^2 in terms of y given the condition that y(0)=1.

a)

$y=\ln\left|1+x^2\right|$

b)

$y=2\ln\left|1+x^2\right|$

c)

$y=e^{2\arctan\left(x\right)}$

d)

$y=e^{\arctan\left(x\right)}$

e)

$y=e^{\arctan\left(2x\right)}$

88.

Which of the following is the solution to the differential equation dy/dx=-x/ye^x^2 with initial condition (0,1)? Assume y>0.

a)

$e^{-x^2}$

b)

$e^{x^2}$

c)

$e^{\frac{-x^2}{2}}$

d)

$e^{\frac{x^2}{2}}$

e)

$-e^{\frac{x^2}{2}}$

89.

Find the general solution of the differential equation y’–ysin(x)=0.

a)

$|y|=Ce^{-\sin\left(x\right)}$

b)

$|y|=Ce^{\sin\left(x\right)}$

c)

$|y|=Ce^{-\cos\left(x\right)}$

d)

$|y|=Ce^{\cos\left(x\right)}$

e)

$|y|=-Ce^{\cos\left(x\right)}$

90.

Find the general solution of the differential equation dy/dx=2x/4+x^2.

a)

$y=\arctan(2x)+C$

b)

$y=\frac{1}{2}\arctan(2x)+C$

c)

$y=\arctan(x^2+4)+C$

d)

$y=\ln|x^2+4|+C$

e)

$y=2x\ln|x^2+4|+C$

91.

Find the general solution of the differential equation dy/dx=3y/7x.

a)

$y^{\frac{1}{3}}=Cx^{\frac{1}{7}}$

b)

$y^3=Cx^7$

c)

$y=Cx^{\frac{1}{7}}$

d)

$y^{\frac{1}{7}}=Cx^3$

e)

$y^7=Cx^{\frac{1}{3}}$

92.

Let y=f(t) be a solution to the differential equation dy/dt=ky, where k is a constant. Values of f for selected values of t are given in the table above. Which of the following is an expression for f(t)?

a)

$4e^{\frac{t}{2}}\ln3$

b)

$e^{\left(\frac{t}{2}\ln9\right)}+3$

c)

$2t^2+4$

d)

$4t+4$

93.

A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have heard the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant?

a)

$\frac{dp}{dt}=kp$

b)

$\frac{dp}{dt}=kp(N–p)$

c)

$\frac{dp}{dt}=kp(p–N)$

d)

$\frac{dp}{dt}=kt(N–t)$

e)

$\frac{dp}{dt}=kt(t–N)$

94.

A puppy weighs 2.0 pounds at birth and 3.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is 3 months old?

a)

4.2 Pounds

b)

4.6 Pounds

c)

4.8 Pounds

d)

5.6 Pounds

e)

6.5 Pounds

95.

Population y grows according to the equation dy/dx=ky, where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is:

a)

5.000

b)

3.322

c)

0.301

d)

0.200

e)

0.069

96.

During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, how many people are infected 12 days after the epidemic is first discovered?

a)

343

b)

1343

c)

1367

d)

1400

e)

2057

97.

Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple?

a)

$\ln\left(\frac{27}{2}\right)$

b)

$\ln\left(\frac{9}{2}\right)$

c)

$\frac{2\ln\left(3\right)}{\ln\left(2\right)}$

d)

$\frac{3\ln\left(3\right)}{\ln\left(2\right)}$

e)

$\frac{\ln\left(3\right)}{\ln\left(2\right)}$

98.

According to Newton’s law of cooling, the rate at which an object cools (or warms) is directly proportional to the temperature difference between the environment and the object itself. If a pot of boiling water (100°C) is left at room temperature (22°C) and after five minutes the water is only 70°C, find its temperature after another 5 minutes.

a)

51°C

b)

52°C

c)

61°C

d)

62°C

e)

63°C

99.

The population P(t) of a species satisfies the logistic differential equation dP/dt=P(2–P/5000),

Where the initial population P(0)=3000 and t is time in years. What is lim(t->infinity)P(t)?

a)

2500

b)

3000

c)

4200

d)

5000

e)

10000

100.

The number of moose in a national park is modeled by the function M that satisfies the logistic differential equation dM/dt=0.6(1–M/200), where t is the time in years and M(0)=50. What is lim_{(t->infinity)}M(t)?

a)

50

b)

200

c)

500

d)

1000

e)

2000

101.

Which of the following is a separable differential equation?

a)

$\frac{\text{d}u}{\text{d}t}+\frac{2u}{t}=8$

b)

$\frac{\text{d}y}{\text{d}x}-6x^2y=9x^2$

c)

$xy'-2y-x^3\cos x=0$

d)

$\frac{d\theta}{dt}+2\theta=\sin\ t$

102.

Solve the differential equation

\frac{dy}{dx}+2=2y

.

a)

$y=1+\sqrt{\frac{4}{-2x-A}}$

b)

$y=\frac{\beta e^x+2}{2}$

c)

$y=1+\alpha e^{2x}$

d)

$y=1+\frac{1}{-4x-A}$

103.

Find the general solution of the differential equation

2y\ dy-\left(y^2+1\right)\cos x\ dx=0

a)

$y=\sqrt{Ae^{\left(\sin x-1\right)}}$

b)

$y=\sqrt{Ae^{\sin x}-1}$

c)

$y=\sqrt{\frac{A}{e^{\sin x}}-1}$

d)

$y=\sqrt{\sin x-1}$

104.

If

y=\frac{e^{\left(\alpha x+\beta\right)}}{\gamma}

is the solution for differential equation

\frac{dy}{dx}=6y

, find the values of

\alpha,

\beta

and

\gamma

when

y=1,\ x=2

.

a)

$\alpha=1,\ \beta=0,\ \gamma=6$

b)

$\alpha=6,\ \beta=12,\ \gamma=6$

c)

$\alpha=12,\ \beta=6,\ \gamma=6$

d)

$\alpha=6,\ \ \beta=-12,\ \gamma=1$

105.

If the gradient of a curve is given as

\frac{1}{x^2y}

, find the equation of the curve at point

\left(2,3\right)

.

a)

$y=\sqrt{\frac{10x-2}{x}}$

b)

$y=\sqrt{9-\ln16-\frac{2}{x}}$

c)

$y=\sqrt{-\frac{1}{x}+\frac{19}{2}}$

106.

Which of the following differential equation CAN NOT be separated?

a)

$\frac{dy}{dx}+1=4x$

b)

$y^2\frac{dy}{dx}=\sqrt{xy}-2x^2\sqrt{y}$

c)

$\frac{dy}{dx}+\frac{2}{x}y=\frac{1}{x+1}$

d)

$\frac{dy}{dx}-e^xy=e^x$

107.

Do you think Week 3 questions are hard?

a)

Yes

b)

No

108.

Given the curve $Ax^2\ +\ y^2=B$ is the solution of the differential equations $\frac{dy}{dx}=-\ \frac{x}{y}$ , at $\left(2,0\right)$ . Find the values of $A$ and $B.$

a)

$A=\frac{1}{2},\ B=\frac{1}{4}$

b)

$A=\frac{1}{2},\ B=1$

c)

$A=1,\ B=4$

d)

$A=-1,\ B=4$

109.

Find the general solution for the differential equation $\frac{dy}{dx}=xy$ .

a)

$y=e^{\frac{x^2}{2}}+c$

b)

$y=\alpha e^{\frac{x^2}{2}}$

c)

$y=Ae^{x^2}$

d)

$y=\sqrt{\frac{1}{x^2+A}}$

110.

Given that the acceleration of an object satisfies the differential equation $e^t\frac{dv}{dt}=2\sqrt{v}$ . Find the equation of the velocity of the object if it started at $v=1.$

a)

$v=\left(2-e^{-t}\right)^2$

b)

$v=\left(3-2e^{-t}\right)^2$

c)

$v=\left(2e^{-t}-1\right)^2$

111.

Given the differential equation $\frac{dP}{dt}=5P$

A) Find the general solution for P .

B)Find the particular solution for P to the differential equation given $P\left(0\right)=418$ .

a)

A) $P=e^{5\alpha t}$
B) $P=e^{418t}$

b)

A) $P=\alpha e^{5t}$
B) $P=418e^{5t}$

c)

A) $P=\beta e^t$
B) $P=418e^t$

d)

A) $P=\alpha e^{10t}$
B) $P=418e^{10t}$

Differential Equations Exam

Total questions: 111

Worksheet time: 1hrs 26mins

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