Free Printable Differential Equation Finals

1.

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

The particular solution is

a)

B

b)

C

c)

D

d)

E

2.

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

a)

Separable.

b)

Non Separable

3.

Is this a Linear Differential equation

a)

Yes

b)

No

c)

Maybe

d)

red

4.

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)

$0=e^x+C$

5.

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

6.

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

7.

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)

$y=zkx$

8.

The degree of a differential equation is define by a

a)

Positive real number

b)

Positive rational number

c)

Positive integer

d)

All the above

9.

Solution of a linear differential equation of first order can be obtain by

a)

multiplying on both the sides by its integrating factor.

b)

adding on both the sides by integrating factor.

c)

subtracting integrating factor from both sides

d)

None of the above

10.

The complete solution of Linear Differential equations involves

a)

complete function + particular integral

b)

complementary function + particular integral

c)

complementary function + definite integral

d)

complete function + indefinite integral

11.

The particular integral of the equation $\left(D-1\right)y=e^{3x}$ is

a)

$\frac{e^{3x}}{2}$

b)

$\frac{e^{-3x}}{2}$

c)

$\frac{e^{3x}}{4}$

d)

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

12.

The C.F. of the equation $\left(D^2-9\right)y=e^{-3x}+1+e^{3x}$ is

a)

$c_1e^{-3x}+c_2e^{-x}$

b)

$c_1e^{3x}+c_2e^{3x}$

c)

$c_1e^{3x}+c_2e^{-x}$

d)

$c_1e^{3x}+c_2e^{-3x}$

13.

Choose the right description for the given differential equation

a)

Ordinary, 2nd order, degree 2, non linear

b)

Partial, 1st order, degree 2, linear.

c)

Ordinary, 1st order, degree 2, linear

d)

Partial, 2nd order, degree 1, non linear

14.

Number of arbitrary constant in the general solution of a differential equation of degree 3 and order 4 is

a)

3

b)

4

c)

0

d)

4³

15.

$\frac{\text{d}^2x}{\text{d}y^2}+6\frac{\text{d}x}{\text{d}y}-5y=0\$

The general solution to the DE is,

a)

$y=Ae^x+Be^{5x}$

b)

$y=Ae^{-x}+Be^{-5x}$

c)

$y=Ae^{\left(-3+\sqrt{14}\right)x}+Be^{\left(-3-\sqrt{14}\right)x}$

d)

$y=A\cos\left(-3+\sqrt{14}\right)x+B\sin\left(-3-\sqrt{14}\right)x$

16.

Find a 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 = x⁴/4

d)

y = ¼(x² - 5)²

17.

Which of the following equations solves dy/dt = ky?

a)

y = y₀e^kt

b)

y = mx + b

c)

y = a(x - h)² + k

d)

y = Asin[B(x - C)] + D

18.

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)

$y=2t^3-16$

19.

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=P₀e^5t

B) P= 5e^418t

b)

A) P=P₀e^5t

B) P= 418e^5t

c)

A) P=P₀e^t

B) P= 418e^t

d)

A) P=P₀e^10t

B) P= 418e^10t

20.

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

21.

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

a)

y = ln (2x²/2 + C) / 2

b)

y = ln(2x²/2)+c

c)

y = e^2x+c

d)

y = mx+b

22.

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)

23.

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

The particular solution is

a)

B

b)

C

c)

D

d)

E

24.

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

25.

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

a)

b)

c)

d)

26.

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)

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

27.

Solve the Initial Value Problem

a)

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

b)

$y=-x^2$

c)

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

d)

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

28.

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)

$0=e^x+C$

29.

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

30.

The solution of linear differential equation $x\frac{\text{d}y}{\text{d}x}+2y=x^2$

a)

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

b)

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

c)

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

d)

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

31.

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)

$\ln y=x+C$

32.

What is the order and degree of the differential equation?

a)

1, 2

b)

2, 1

c)

3, 2

d)

2, 3

33.

What is the order of a differential equation?

a)

the highest power of y

b)

The highest power of x

c)

the highest order of the derivative

d)

the coefficient of y

34.

Example of homogenous equation

a)

m²+2m+3

b)

m²+3m+2

c)

(m²+5m+6)y=0

d)

m²+5m+6=5

35.

Identify the roots of the equation m²-8m+16=0 is

a)

Real and distinct

b)

imaginary

c)

real and equal

d)

no roots

36.

It is a type of differential equation wherein every term has the same degree.

a)

Variable Separable

b)

Exact D.E

c)

Non-exact D.E

d)

Homogenous D.E

37.

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

a)

y' = y/2x

b)

y' = 2y - x

c)

y'' = 2x

d)

y'-2y = 0

38.

The particular integral of the equation $\left(D-1\right)y=e^{3x}$ is

a)

$\frac{e^{3x}}{2}$

b)

$\frac{e^{-3x}}{2}$

c)

$\frac{e^{3x}}{4}$

d)

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

39.

The particular integral of the equation $\left(D^3-1\right)y=\left(e^x+1\right)^2$ is

a)

$\frac{e^{-2x}}{7}+\frac{2x}{3}-1$

b)

$\frac{e^{3x}}{7}+\frac{2x}{3}-1$

c)

$\frac{e^{2x}}{7}+\frac{5x}{3}-1$

d)

$\frac{e^{2x}}{7}+\frac{2x}{3}-1$

40.

it is number of the highest degree in a differential equation

a)

variable

b)

degree

c)

order

d)

derivative

41.

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

42.

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

43.

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

a)

x

b)

y

c)

d

d)

secret

44.

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

a)

x

b)

y

c)

d

d)

none of the options syempre!

45.

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)

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

46.

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)

$\sin y=0+C$

47.

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)

48.

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

49.

The general solution of the DE $\frac{d^2y}{dx^{2}}+4\frac{dy}{dx}−5y=0$

a)

$y=Ae^x+Be^{-5x}$

b)

$y=Ae^x+Be^{5x}$

c)

$y=Ae^{-x}+Be^{-5x}$

d)

$y=Ae^{-x}+Be^{5x}$

50.

What is the order and degree of the differential equation?

a)

1, 2

b)

2, 1

c)

3, 2

d)

2, 3

Differential Equation Finals

Total questions: 50

Worksheet time: 1hrs 2mins

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