# MCQ in Differential Equations Part 1 | Mathematics Board Exam

(Last Updated On: June 8, 2023) This is the Multiple Choice Questions Part 1 of the Series in Differential Equations topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

• MCQ in types of Differential Equations | MCQ in Order of Differential Equations | MCQs in Degree of Differential Equations | MCQ in types of solutions of Differential Equations | MCQ in Applications of Differential Equations

#### Start Practice Exam Test Questions Part 1 of the Series

Choose the letter of the best answer in each questions.

Problem 1:

Determine the order and degree of the differential equation, A. Fourth order, first degree

B. Third order, first degree

C. First order, fourth degree

D. First order, third degree.

Solution: Review Solution for Number 1

Problem 2:

Which of the following equations is an exact DE?

A. (x2 + 1) dx – xy dy = 0

B. x dy + (3x – 2y) dx = 0

C. 2xy dx + (2 + x2) dy = 0

D. x2y dy – y dx = 0

Solution: Review Solution for Number 2

Problem 3:

Which of the following equations is a variable separable DE?

A. (x + x2 y) dy = (2x + xy2) dx

B. (x + y) dx – 2y dy = 0

C. 2y dx = (x2 + 1) dy

D. y2 dx + (2x – 3y) dy = 0

Solution: Review Solution for Number 3

Problem 4: ECE Board April 1998

The equation y2 = cx is general solution of:

A. y’ = 2y / x

B. y’ = 2x / y

C. y’ = y / 2x

D. y’ = x / 2y

Solution: Review Solution for Number 4

Problem 5: EE Board March 1998

Solve the differential equation: x(y – 1) dx + (x + 1) dy = 0. If y = 2 when x = 1.

A. 1.80

B. 1.48

C. 1.55

D. 1.63

Solution: Review Solution for Number 5

Problem 6: EE Board October 1997

If dy = x2 dx; what is the equation of y in terms of x if the curve passes through (1, 1).

A. x2 – 3y + 3 = 0

B. x3 – 3y + 2 = 0

C. x3 + 3y2 + 2 = 0

D. 2y + x3 + 2 = 0

Solution: Review Solution for Number 6

Problem 7: ECE Board November 1998

Find the equation of the curve at every point of which the tangent line has a slope of 2x.

A. x = -y2 + C

B. y = -x2 + C

C. y = x2 + C

D. x = y2 + C

Solution: Review Solution for Number 7

Problem 8: ECE Board April 1995

Solve (cox x cos y – cotx) dx – sin x sin y dy = 0

A. sin x cos y = ln (c cos x)

B. sin x cos y = ln (c sin x)

C. sin x cos y = -ln (c sin x)

D. sin x cos y = -ln (c cos x)

Solution: Review Solution for Number 8

Problem 9: EE Board October 1997

Solve the differential equation dy – x dx = 0, if the curve passes through (1, 0)?

A. 3x2 + 2y – 3 = 0

B. 2y2 + x2 – 1 = 0

C. x2 – 2y – 1 = 0

D. 2x2 + 2y – 2 = 0

Solution: Review Solution for Number 9

Problem 10: ME Board April 1996

What is the solution of the first order differential equation y(k + 1) = y(k) + 5.

A. y(k) = 4 – 5/k

B. y(k) = 20 + 5k

C. y(k) = C – k, where C is constant

D. The solution is non-existence for real values of y.

Solution: Review Solution for Number 10

Problem 11: EE Board April 1995

Solve (y – √(x2 + y2)) dx – x dy = 0

A. √(x2 + y2 ) + y = C

B. √(x2 + y2 + y) = C

C. √(x + y) + y = C

D. √(x2 – y) + y = C

Solution: Review Solution for Number 11

Problem 12: ECE Board November 1994

Find the differential equation whose general solution is y = C1x + C2ex.

A. (x – 1) y” – xy’ + y = 0

B. (x + 1) y” – xy + y = 0

C. (x – 1) y” + xy’ + y = 0

D. (x + 1) y” + xy’ + y = 0

Solution: Review Solution for Number 12

Problem 13: EE Board October 1995

Find the general solution of y’ = y sec x

A. y = C (sec x + tan x)

B. y = C (sec x – tan x)

C. y = C (sec x tan x)

D. y = C (sec2 x + tan x)

Solution: Review Solution for Number 13

Problem 14: EE Board April 1996

Solve xy’ (2y – 1) = y (1 – x)

A. ln (xy) = 2 (x – y) + C

B. ln (xy) = x – 2y + C

C. ln (xy) = 2y – x + C

D. ln (xy) = x + 2y + C

Solution: Review Solution for Number 14

Problem 15: EE Board April 1996

Solve (x + y) dy = (x – y) dx

A. x2 + y2 = C

B. x2 + 2xy + y2 = C

C. x2 – 2xy – y2 = C

D. x2 – 2xy + y2 = C

Solution: Review Solution for Number 15

Problem 16:

Solve the linear equation: dy/dx + y / x = x2

A. xy2 = x3 / 4 + C

B. xy = x4 / 4 + C

C. x2y = x4 / 4 + C

D. y = x3 / 4 + C

Solution: Review Solution for Number 16

Problem 17: CE Board May 1997

Find the differential equations of the family of lines passing through the origin.

A. y dx – x dy = 0

B. x dy – y dx = 0

C. x dx + y dy = 0

D. y dx + x dy = 0

Solution: Review Solution for Number 17

Problem 18: CE Board May 1996

What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis.

A. 2x dy – y dx = 0

B. x dy + y dx = 0

C. 2y dx – x dy = 0

D. dy/dx – x = 0

Solution: Review Solution for Number 18

Problem 19: CE Board November 1995

Determine the differential equation of the family of lines passing through (h, k).

A. (y – k) dx – (x – h) dy = 0

B. (y – h) + (y – k) = dy/dx

C. (x – h) dx – (y – k) dy = 0

D. (x + h) dx – (y – k) dy = 0

Solution: Review Solution for Number 19

Problem 20:

Determine the differential equation of the family of circles with center on the y-axis.

A. (y”)3 – xy” + y’ = 0

B. y” – xyy” + y’ = 0

C. xy” – (y’)3 – y’ = 0

D. (y’)3 + (y”)2 + xy = 0

Solution: Review Solution for Number 20

Problem 21: EE Board April 1997

Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years?

A. 88.60

B. 95.32

C. 92.16

D. 90.72

Solution: Review Solution for Number 21

Problem 22:

The population of a country doubles in 50 years. How many years will it be five times as much? Assume that the rate of increase is proportional to the number inhabitants.

A. 100 years

B. 116 years

C. 120 years

D. 98 years

Solution: Review Solution for Number 22

Problem 23:

Radium decomposes at a rate proportional to the amount present. If the half of the original amount disappears after 1000 years, what is the percentage lost in 100 years?

A. 6.70%

B. 4.50%

C. 5.35%

D. 4.30%

Solution: Review Solution for Number 23

Problem 24: ECE Board November 1998

Find the equation of the family of orthogonal trajectories of the system of parabolas y2 = 2x + C.

A. y = Ce-x

B. y = Ce2x

C. y = Cex

D. y = Ce-2x

Solution: Review Solution for Number 24

Problem 25:

According to Newton’s law of cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperatures of the substance and that of air. If the temperature of the air is 30° and the substance cools from 100° to 70° in 15 minutes, how long will it take to cool 100° to 50°?

A. 33. 59 min

B. 43.60 min

C. 35.39 min

D. 45.30 min

Solution: Review Solution for Number 25

Problem 26:

An object falls from rest in a medium offering a resistance. The velocity of the object before the object reaches the ground is given by the differential equation dV/dt + V/10 = 32, ft/sec. What is the velocity of the object one second after if falls?

A. 40.54 ft/sec

B. 38.65 ft/sec

C. 30.45 ft/sec

D. 34.12 ft/sec

Solution: Review Solution for Number 26

Problem 27:

In a tank are 100 liters of brine containing 50 kg. total of dissolved salt. Pure water is allowed to run into the tank at the rate of 3 liters a minute. Brine runs out of the tank at the rate of 2 liters a minute. The instantaneous concentration in the tank is kept uniform by stirring. How much salt is in the tank at the end of one hour?

A. 15.45 kg

B. 19.53 kg

C. 12.62 kg

D. 20.62 kg

Solution: Review Solution for Number 27

Problem 28:

A tank initially holds 100 gallons of salt solution in which 50 lbs of salt has been dissolved. A pipe fills the tank with brine at the rate of 3 gpm, containing 2 lbs of dissolved salt per gallon. Assuming that the mixture is kept uniform by stirring, a drain pipe draws out of the tank the mixture at 2 gpm. Find the amount of salt in the tank at the end of 30 minutes.

A. 171.24 lbs

B. 124.11 lbs

C. 143.25 lbs

D. 105.12 lbs

Solution: Review Solution for Number 28

Problem 29: ME Board April 1998

If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuous compounded account?

A. P5,750

B. P6,750

C. P7,500

D. P6,350

Solution: Review Solution for Number 29

Problem 30: ME Board October 1997

A nominal interest of 3% compounded continuously is given on the account. What is accumulated amount of P10,000 after 10 years.

A. P13,620.10

B. P13,500.10

C. P13,650.20

D. P13,498.60

Solution: Review Solution for Number 30

#### Online Question and Answer in Differential Equations Series

Following is the list of multiple choice questions in this brand new series:

MCQ in Differential Equations
PART 1: MCQ from Number 1 – 50                                 Answer key: PART 1
PART 2: MCQ from Number 51 – 100                            Answer key: PART 2

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