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.

### Multiple Choice Questions Topic Outline

- MCQs in types of Differential Equations | MCQs in Order of Differential Equations | MCQs in Degree of Differential Equations | MCQs in types of solutions of Differential Equations | MCQs in Applications of Differential Equations

### Online Questions and Answers in Differential Equations Series

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

**Differential Equations MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

**MCQs from Number 51 – 100**Answer key:

**PART II**

### Start Practice Exam Test Questions Part I 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.

**Problem 2: **

Which of the following equations is an exact DE?

- A. (x
^{2}+ 1) dx – xy dy = 0 - B. x dy + (3x – 2y) dx = 0
- C. 2xy dx + (2 + x
^{2}) dy = 0 - D. x
^{2}y dy – y dx = 0

**Problem 3: **

Which of the following equations is a variable separable DE?

- A. (x + x
^{2}y) dy = (2x + xy^{2}) dx - B. (x + y) dx – 2y dy = 0
- C. 2y dx = (x
^{2}+ 1) dy - D. y
^{2 }dx + (2x – 3y) dy = 0

**Problem 4: ECE Board April 1998**

The equation y^{2 }= cx is general solution of:

- A. y’ = 2y / x
- B. y’ = 2x / y
- C. y’ = y / 2x
- D. y’ = x / 2y

**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

**Problem 6: EE Board October 1997**

If dy = x^{2 }dx; what is the equation of y in terms of x if the curve passes through (1, 1).

- A. x
^{2}– 3y + 3 = 0 - B. x
^{3}– 3y + 2 = 0 - C. x
^{3}+ 3y^{2 }+ 2 = 0 - D. 2y + x
^{3}+ 2 = 0

**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 = -y
^{2}+ C - B. y = -x
^{2 }+ C - C. y = x
^{2}+ C - D. x = y
^{2}+ C

**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)

**Problem 9: EE Board October 1997**

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

- A. 3x
^{2}+ 2y – 3 = 0 - B. 2y
^{2}+ x^{2}– 1 = 0 - C. x
^{2}– 2y – 1 = 0 - D. 2x
^{2}+ 2y – 2 = 0

**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.

**Problem 11: EE Board April 1995**

Solve (y – √(x^{2} + y^{2})) dx – x dy = 0

- A. √(x
^{2}+ y^{2 }) + y = C - B. √(x
^{2}+ y^{2}+ y) = C - C. √(x + y) + y = C
- D. √(x
^{2}– y) + y = C

**Problem 12: ECE Board November 1994**

Find the differential equation whose general solution is y = C_{1}x + C_{2}e^{x}.

- 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

**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 (sec
^{2}x + tan x)

**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

**Problem 15: EE Board April 1996**

Solve (x + y) dy = (x – y) dx

- A. x
^{2}+ y^{2}= C - B. x
^{2}+ 2xy + y^{2}= C - C. x
^{2}– 2xy – y^{2 }= C - D. x
^{2}– 2xy + y^{2 }= C

**Problem 16: **

Solve the linear equation: dy / dx + y / x = x^{2}

- A. xy
^{2}= x^{3}/ 4 + C - B. xy = x
^{4}/ 4 + C - C. x
^{2}y = x^{4}/ 4 + C - D. y = x
^{3}/ 4 + C

**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

**Problem 18: CE Board May 1996**

What is the differentia 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

**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

**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

**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

**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

**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%

**Problem 24: ECE Board November 1998**

Find the equation of the family of orthogonal trajectories of the system of parabolas y^{2} = 2x + C.

- A. y = Ce
^{-x} - B. y = Ce
^{2x} - C. y = Ce
^{x} - D. y = Ce
^{-2x}

**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

**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

**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

**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

**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

**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

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