DIFFERENTIAL EQUATIONS 1001

Question 1

Determine the order and degree of the differential equation
2x (d^4y/dy^4) + 5x2 (dy/dx)^3 - xy = 0

A. Fourth order, first degree
B. Third order, first degree
C. First order, fourth degree
D. First order, third degree

Answer 1

A. Fourth order, first degree

Question 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’y dy - y dx = 0

Answer 2

C. 2xy dx + (2 + x^2) dy = 0

Question 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

Answer 3

C. 2y dx = (x^2 + 1) dy

Question 4

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

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

Answer 4

C. y’ = y/2x

Question 5

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

A. 1.80
B. 1.48
C. 1.55
D. 1.63

Answer 5

C. 1.55

Question 6

If dy = x^2dx; 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

Answer 6

B. x^3 - 3y + 2 = 0

Question 7

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

Answer 7

C. y = x^2 + C

Question 8

Solve (cos x cos y - cot x) dx - sin x sin y dy = 0

A. sin x cos y = In (c cos x)
B. sin x cos y = In (c sin x)
C. sin x cos y = - In (c sin x)
D. sin x cos y = - In (c cos x)

Answer 8

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

Question 9

Solve the differential equation dy -xdx = 0, if the curve passes through (1,0)?

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

Answer 9

C. x^2 - 2y - 1 = 0

Question 10

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-existent for real values of y

Answer 10

B. y(k) = 20 + 5k

Question 11

Solve (y - √(x2 + y2) ) dx - xdy = 0

A. √(x2+ y2) + y = C
B. √(x2 + y2 + y) = C
C. √(x + y) + y = C
D. √(x2 - y) + y = C

Answer 11

A. √(x2+ y2) + y = C

Question 12

Find the differential equation whose general solution is
y = C1^x + C2e^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

Answer 12

A. (x-1) y” - xy’ + y = 0

Question 13

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)

Answer 13

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

Question 14

Solve xy’ (2y-1) = y (1-x)

A. In (xy)=2 (x-y) + C
B. In (xy) = x - 2y + C
C. In (xy) = 2y - x + C
D. In (xy) = x + 2y + C

Answer 14

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

Question 15

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

Answer 15

C. x^2 - 2xy - y^2 = C

Question 16

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

A. xy2 = x^2 / 4 + C
B. xy = x^4 / 4 +C
C. x^2 y = x^4 / 4 + C
D. y = x^3 / 4 +C

Answer 16

B. xy = x^4 / 4 +C

Question 17

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

A. ydx – xdy = 0
B. xdy - ydx = 0
C. xdx + ydy = 0
D. ydx + xdy = 0

Answer 17

B. xdy - ydx = 0

Question 18

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

A. 2xdy - ydx = 0
B. xdy + ydx = 0
C. 2ydx - xdy = 0
D. dy/dx - x = 0

Answer 18

A. 2xdy - ydx = 0

Question 19

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

Answer 19

A. (y-k)dx - (x-h)dy = 0

Question 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’ = 0
C. xy” - (y’)^3 - y’ = 0
D. (y’)^3 + (y”)^2 + xy = 0

Answer 20

C. xy” - (y’)^3 - y’ = 0

Question 21

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

Answer 21

C. 92.16

Question 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 of inhabitants.

A. 100 years
B. 116 years
C. 120 years
D. 98 years

Answer 22

B. 116 years

Question 23

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

A. 6.70%
B. 4.50%
C. 5.36%
D. 4.30%

Answer 23

A. 6.70%

Question 24

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

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

Answer 24

A. y = Ce^-x

Question 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 temperature 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. 43.50 min C. 35.39 min D. 45.30 min

Answer 25

A. 33.59 min.

Question 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 it falls? A. 40.54 B. 38.65 C. 30.45 D. 34.12

Answer 26

C. 30.45

Question 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.

Answer 27

B. 19.53 kg.

Question 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.

Answer 28

A. 171.24 lbs.

Question 29

If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuously compounded account? A. P5,750 B. P6,750 C. P7,500 D. P6,350

Answer 29

B. P6,750

Question 30

A nominal interest of 3% compounded continuously is given on the account. What is the accumulated amount of P10,000 after 10 years? A. P13,620.10 B. P13,500.10 C. P13,650.20 D. P13,498.60

Answer 30

D. P13,498.60