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1.

What is the order of the differential equation $y''+xy'+x^2y=\sin x$ ?

a)

4

b)

3

c)

2

d)

1

2.

Which differential equation has the solution $y=\sin x$ ?

a)

$y+y''=0$

b)

$y-y''=0$

c)

$y+y'-y''=0$

d)

$y-y'+y''=0$

3.

Which function is the general solution of the differential equation $y'''+y''+y'+y=0$ ?

a)

$y=A\sin x+Be^{-x}$

b)

$y=A\sin2x-Be^{-x}$

c)

$y=A\sin x+Be^{-2x}$

d)

$y=A\sin2x-Be^{-2x}$

4.

Using the separation of variables, what is the solution of the differential equation $\frac{dy}{dx}=\frac{3xy}{1-x^2}$ ?

a)

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

b)

$y=\frac{K}{\left(1-x^2\right)^{\frac{3}{2}}}$

c)

$y=\frac{K}{\left(1-x^2\right)^{\frac{5}{2}}}$

d)

$y=\frac{K}{\left(\frac{3}{2}-x^2\right)^{\frac{5}{2}}}$

5.

Using the separation of variables, what is the solution of the differential equation $y'=-\frac{x}{y}$ , given the initial condition $y\left(1\right)=4$ ?

a)

$y^2=14+2x^2$

b)

$y^2=18-2x^2$

c)

$y^2=15+x^2$

d)

$y^2=17-x^2$

6.

Let the function $y=f\left(x\right)$ be a solution of $\frac{dy}{dx}=4-x$ , where $y\left(-1\right)=2.$

What is the solution?

a)

$y=2x+\frac{x^2}{2}-\frac{11}{2}$

b)

$y=2x-\frac{x^2}{2}-\frac{11}{2}$

c)

$y=4x-\frac{x^2}{2}+\frac{13}{2}$

d)

$y=-4x+\frac{x^2}{2}-\frac{13}{2}$

7.

Consider the differential equation $\frac{dy}{dx}=\frac{y}{x}$ .

Which of the following equations best represents this relationship between x and y ?

a)

$y^2=x^2+c$

b)

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

c)

$y=x\ln\left|y\right|+c$

d)

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

8.

The equation $y=e^{ax}$ satisfies the differential equation $\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0.$

What are the possible values of $a\ ?$

a)

$a=-2$ or $a=3$

b)

$a=-1$ or $a=6$

c)

$a=2$ or $a=-3$

d)

$a=1$ or $a=-6$

9.

Which of the following differential equations can NOT be solved by separation of the variables?

a)

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

b)

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

c)

$\frac{dy}{dx}=xy$

d)

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

10.

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)

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

11.

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)

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

12.

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)

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

13.

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

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

14.

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)

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

15.

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)

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

16.

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

17.

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)

18.

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

19.

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

20.

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

21.

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

The particular solution is

a)

B

b)

C

c)

D

d)

E

22.

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.

23.

Find f(1) given f'(x)

a)

-12

b)

0

c)

14

d)

20

24.

Evaluate:

a)

4

b)

6

c)

8

d)

10

25.

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)

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

26.

Solve the Initial Value Problem

a)

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

b)

$y=-x^2$

c)

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

d)

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

27.

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

28.

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

29.

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

30.

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

31.

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

is an ordinary differential equation

a)

True

b)

False

32.

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

33.

it is number of the highest degree in a differential equation

a)

variable

b)

degree

c)

order

d)

derivative

34.

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

35.

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

36.

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

37.

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

a)

True

b)

False

38.

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

a)

x

b)

y

c)

d

d)

secret

39.

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

a)

x

b)

y

c)

d

d)

none of the options syempre!

40.

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

a)

Yes

b)

No

c)

neither

d)

Either

41.

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)

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

42.

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)

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

43.

A Differential Equation is an equation that relates to an unknown function and one or more of its derivatives.

a)

TRUE

b)

FALSE

44.

The____ of a differential equation is the order of the _____ that occurs in the equation.

a)

order, lowest derivative

b)

order, Highest derivative

45.

Consider the First-Order DE $\frac{dy}{dx}=2x$ What is the general solution?

a)

$y=x^2$

b)

$y=x^2+C$

c)

$y=C$

d)

$y=x^2-C$

46.

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)

$y=2t^3-16$

47.

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)

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

48.

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)

$\sin y=0+C$

49.

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)

$0=e^x+C$

50.

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)

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

51.

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)

52.

a)

A

b)

B

c)

C

d)

D

e)

E

53.

A partial differential equation requires

a)

exactly one independent variable

b)

two or more independent variables

c)

more than one dependent variable

d)

equal number of dependent and independent variables

54.

Using substitution, which of the following equations are solutions to the partial differential equation?

a)

b)

c)

d)

55.

What is the order & degree of the p.d.e $x^2\left(\frac{\partial z}{\partial x}\right)^2=z\left(x-y\frac{\partial z}{\partial y}\right)$

a)

1 & 2

b)

2 & 1

c)

2 & 2

d)

none of the above

56.

Singular solution of Partial differential equation can be obtained by

a)

General solution

b)

Complete solution

c)

both a & b

d)

none of the above

57.

If u is homogeneous function of order n then $\frac{\partial u}{\partial x}\ ,\ \frac{\partial u}{\partial y}$ both are homogeneous function of order

a)

n

b)

n-1

c)

n+1

d)

n-2

58.

If $f\left(x,y\right)=0\$ then what is the value of $\frac{dy}{dx}$ is

a)

$\frac{f_x}{f_y}$

b)

$-\frac{f_x}{f_y}$

c)

$\frac{f_y}{f_x}$

d)

$-\frac{f_y}{f_x}$

59.

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)

$e^{\int Qdx_{ }}$

60.

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

a)

Separable.

b)

Non Separable.

61.

What is homogeneous

a)

same degree

b)

same order

c)

different degree

d)

different order

62.

What do mean by linear

a)

Degree 1

b)

Degree 2

c)

Degree 3

d)

Degree 4

63.

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

a)

Separable.

b)

Non Separable

64.

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

a)

Separable.

b)

Non Separable.

65.

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

a)

Separable.

b)

Non Separable.

66.

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

a)

Separable.

b)

Non Separable.

67.

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

a)

Separable.

b)

Non Separable.

68.

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

a)

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

b)

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

c)

$x^2=y^2+C$

d)

$y^2=x^2+2C$

69.

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

a)

$y=Ax$

b)

$y^2=\ln x+C$

c)

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

d)

$A=yx$

70.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=3y$

a)

$y=Ce^{3x}$

b)

$y=3x+C$

c)

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

d)

$\ln y=x+C$

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.

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)

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

74.

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)

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

75.

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)

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

76.

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)

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

77.

What is the order in the given differential equation:

(1-x) y'' - 4xy' + 5y = cosx

a)

1

b)

2

c)

3

d)

4

78.

The equation y^2 = Cx is general solution of?

a)

y' = y/2x

b)

y' = 2y - x

c)

y'' = 2x

d)

y'-2y = 0

79.

What is the order & degree of the p.d.e $x^2\left(\frac{\partial z}{\partial x}\right)^2=z\left(x-y\frac{\partial z}{\partial y}\right)$

a)

1 & 2

b)

2 & 1

c)

2 & 2

d)

none of the above

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Total questions: 79

Worksheet time: 53mins

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