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Day 3 Board Exam > DIFFERENTIAL EQUATIONS 1001 > Flashcards

DIFFERENTIAL EQUATIONS 1001 Flashcards

1
Q

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

A

A. Fourth order, first degree

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2
Q

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

A

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

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3
Q

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

A

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

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4
Q

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

A

C. y’ = y/2x

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5
Q

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

A

C. 1.55

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6
Q

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

A

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

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7
Q

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

A

C. y = x^2 + C

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8
Q

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)

A

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

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9
Q

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

A

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

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10
Q

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

A

B. y(k) = 20 + 5k

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11
Q

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

A

A. √(x2+ y2) + y = C

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12
Q

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

A

A. (x-1) y” - xy’ + y = 0

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13
Q

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)

A

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

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14
Q

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

A

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

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15
Q

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

A

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

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16
Q

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

A

B. xy = x^4 / 4 +C

17
Q

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

A

B. xdy - ydx = 0

18
Q

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

A

A. 2xdy - ydx = 0

19
Q

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

A

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

20
Q

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

A

C. xy” - (y’)^3 - y’ = 0

21
Q

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

A

C. 92.16

22
Q

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

A

B. 116 years

23
Q

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%

A

A. 6.70%

24
Q

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

A

A. y = Ce^-x

25
Q

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

A

A. 33.59 min.

26
Q

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

A

C. 30.45

27
Q

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.

A

B. 19.53 kg.

28
Q

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.

A

A. 171.24 lbs.

29
Q

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

A

B. P6,750

30
Q

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

A

D. P13,498.60

Day 3 Board Exam (2 decks)

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