Question 1

Solve y' = ySecx,

Accepted Answer

y = C ( Secx + Tanx)

Answer

y = C ( Secx - Tanx)

Answer

y = C ( Cosx + Sinx)

Answer

y = C ( Sinx + Cosx)

Question 2

Find the family of curves with slope 2x.

Accepted Answer

y = x^2 + C

Answer

x = y^2 + C

Answer

x = 2y + C

Answer

y = 2x + C

Question 3

Which of the following is the DE for y = C1 x + C2 e^x.

Accepted Answer

(x-1) y'' - xy' + y = 0

Answer

(x-1) y'' + xy' + y = 0

Answer

(x-1) y'' - xy' - y = 0

Answer

(x-1) y'' + xy' - y = 0

Question 4

If f ' (x) = -f(x), f(1) = 1 then f(x) is,

Accepted Answer

e^1-x

Answer

e^-x

Answer

–e^x

Answer

e^-x-1

Question 5

Find the differential equation for the family of parabolas with vertex at the origin and focus on the x axis.

Accepted Answer

y'= y/2x

Answer

y' = x/2y

Answer

y' = 2y/x

Answer

y’ = 2x/y

Question 6

Find te differential equation of the family of lines that passes through the origin.

Accepted Answer

ydx - xdy = 0

Answer

xdx -ydy=0

Answer

xdx + ydy = 0

Answer

ydx + xdy = 0

Question 7

The population of a certain bacteria grows at the rate which s proportional to the amount present. It doubles in 2 yrs. If in 3 yrs there are 20000 of bacteria present, how many is present initially?

Accepted Answer

7071

Answer

8071

Answer

9071

Answer

10071

Question 8

The time rate of change of a rabbit population P is proportional to the square root of P. At time t = 0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many rabbits will there be one year later?

Accepted Answer

484

Answer

474

Answer

464

Answer

454

Question 9

According to Newton's law of cooling, the rate at which the 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°C and the substance cools from 100°C to 70°C in 15 minutes, how long will it takes in minutes to cool from 100°C to 50°C?

Accepted Answer

33.6

Answer

43.5

Answer

45.3

Answer

35.6

Question 10

In a room temperature of 25°C, a steel ball at 120°C cools down to 80°C in 20 minutes. Find its temperature after one half hour.

Accepted Answer

66.86°C

Answer

56.85°C

Answer

76.85°C

Answer

46.85°C

Question 11

An object falls from rest. If the velocity of the object before it reached the ground is represented by the differential equation dv/dt plus v per 10 equals 32, ft/sec^2, find the velocity of the object in ft/sec after one second.

Accepted Answer

30.45

Answer

34.12

Answer

36.85

Answer

40.54

Question 12

The rate of virus spread is jointly proportional to those infected and uninfected people. If there are 5000 inhabitants at the start of the year and after a week there were 160 people infected, after 2 weeks there are 1200 people infected, how many days will there be for 1500 people to be infected?

Accepted Answer

15days

Answer

20days

Answer

18days

Answer

23days

Question 13

A 400gal tank initially contains 100gal of brine containing 50lb of salt. Brine containing llb of salt per gallon enters the tank at the rate of 5gal/s and well mixed in the tank flows out at the rate of 3gal/s. How much salt will the tank contain when it is full of brine?

Accepted Answer

393.75lb

Answer

373.75lb

Answer

383.75lb

Answer

363.75lb

Question 14

Find the general solution to (D^4-1)y = 0.

Accepted Answer

y=C1e^x + C2e^-x + C3cosx + C4sinx

Answer

y = C1e^x + C2e^-x

Answer

y = C1e^x + C2 xe^-x

Answer

y = C1e^x + C2e^-x + C3xe^x + C4xe^-x

Question 15

Find the general solution to (D^2-D + 2)y = 0.

Accepted Answer

y = e^0.5x (C1cos 1.323x + C2sin1.323x)

Answer

y = e^-0.5x (C1cos 1.323x + C2sin1.323x)

Answer

y = e^x (C1cos 1.323x + C2sin1.323x)

Answer

y = e^0.5x ( C1cos 1.323x – C2sin1.323x)

Question 16

Find the partial integral or partial solution to (D^2 + 1) y = sec x.

Accepted Answer

x sin x + cos x In (cos x)

Answer

sin x - cos x In (cos x)

Answer

sin x + cos x In (cos x)

Answer

x sin x - cos x In (cos x)

Question 17

Solve y'' – 4y' + 3y = Sinx.

Accepted Answer

C₁ e ^3x + C₂ e ^x + 1/5 Cosx + 1/10 Sinx

Answer

C₁ e ^3x + C₂ e ^x– 1/5 Cosx + 1/10 Sinx

Answer

C₁ e ^3x + C₂ e^x – 1/5 Cosx - 1/10 Sinx

Answer

C₁ e ^3x + C₂e ^x+ 1/5 Cosx - 1/10 Sinx

Question 18

The intensity of light at a depth x meters below the surface of the lake satisfies the differential equation dI/dx = -1.4 I. At what depth is the intensity half the intensity lo at the surface where x = 0?

Accepted Answer

0.495m

Answer

0.459m

Answer

0.945m

Answer

0.549m

Question 19

Which of the following differential equations is exact?

Accepted Answer

2xydx + (2 + x^2)dy = 0

Answer

xdy + (3x – 2y)dx = 0

Answer

(x^2 + 1)dx - xydy= 0

Answer

x^2ydx - ydy = 0

Question 20

Find the solution to initial value problem y'' + y' – 2y = -4 , y (0) = 2 and y'(0) = 3.

Accepted Answer

2 + e^x – e ^-2x

Answer

2 + e^x + e^2x

Answer

2 + e^-x – e ^-2x

Answer

2 - e^x – e^-2x

Question 21

Solve the DE y'' + 10y' + 41y = 0

Accepted Answer

y = e^-5x (C₁ Cos4x + C₂ Sin4x)

Answer

y = e^5x (C₁ Cos4x + C₂ Sin4x)

Answer

y = e^4x (C₁ Cos5x + C₂Sin5x)

Answer

y = e^-4x (C₁ Cos5x + C₂ Sin5x)

Question 22

Given is x(y-1) dx – (x+1) dy = 0 and x = 4 when y = 2, find y when x = 2

Accepted Answer

1.23

Answer

2.13

Answer

3.21

Answer

1.32

Question 23

Solve the DE dx/dt = 1 – x – t + xt.

Accepted Answer

2ln(x – 1) – (1 – t)^2 = C

Answer

2ln(x – 1) + (1 – t)^2 = C

Answer

ln(x – 1) + (1 – t)^2 = C

Answer

ln(x – 1) – (1 – t)^2 = C

Question 24

Find the orthogonal trajectories to the family of parabolas y^2 = 2x + C

Accepted Answer

y = Ce^-x

Answer

y = Ce^x

Answer

y = Ce^2x

Answer

y = Ce^-2x

Question 25

Radium decomposed at the rate proportional to the amount present. In 100 yrs, 100mg becomes 96mg. What is the percentage remaining after 100 yrs?

Accepted Answer

92.16%

Answer

91.26%

Answer

96.12%

Answer

92.61%

Question 26

Suppose that at time t = 0. 10,000 people in the city with population of 100,000 people have heard a certain rumor. After one week, the number of those who have heard it has increased to 20,000. Assuming that is satisfies a logistic equation, when will 80% of the city's population have heard the rumor?

Accepted Answer

4.42wks

Answer

4.22wks

Answer

4.12wks

Answer

4.32wks

Question 27

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

Accepted Answer

y(k) = 20 + 5k

Answer

y(k) = C-k, C is a constant

Answer

y(k) = 5k + (1-5^k) / -4

Answer

y(k) = 4 - 5/k

Question 28

Find the particular solution to dx/dt = x/2 if x(0) = 1.

Accepted Answer

x^2 = e^t

Answer

x^4 = t e^t

Answer

x = e^t

Answer

x^3 = e^t

Question 29

Find the solution to dy/dt = ky with initial condition y = 1 when t = 0.

Accepted Answer

y = e^kt

Answer

y = Sinkt

Answer

y^2 = 2t + k

Answer

y = lnkt

Question 30

Given y = e^mx what values of m (-infinity to +infinity) will satisfy the relationship 6y'' - y' – y = 0.

Accepted Answer

-1/3, ½

Answer

1/3, - ½

Answer

1/3, 1/2

Answer

-1/3, -1/2

Question 31

Solve the equation y’ = y/2x.

Accepted Answer

y^2 = cx

Answer

y = cx

Answer

y^2 = cx^3

Answer

y = cx^2

Question 32

A steel ball at 120 deg C cools in 20 minutes to deg C in a room at 25 deg C. Find the temperature of the ball after half an hour

Accepted Answer

66.85 deg C

Answer

55.86 deg C

Answer

45.98 deg C

Answer

40.96 deg C

Question 33

Find the equation of the family of curves at every point of which the tangent line has a slope of 2y.

Accepted Answer

y = Ce^2x

Answer

x = Ca^y

Answer

x = Ce^2y

Answer

y = Ce^x

Question 34

A body moving in a straight line has an acceleration equal to 6t^2, where the time (t) is measured in seconds and distance (s) is measured in feet. If the body starts from rest, how far will it move during the first 2 sec?

Accepted Answer

8 ft

Answer

6 ft

Answer

4 ft

Answer

16 ft

Question 35

A motorboat weighs 32000 lb and its motor provides a thrust of 5000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. Then 1000 dv / dt = 5000 – 100v. If the boat starts from rest, what is the maximum velocity that it can attain?

Accepted Answer

50 ft/s

Answer

40 ft/s

Answer

25 ft/s

Answer

20 ft/s

Question 36

A weight attached to a spring moves up and down, so that the equation of motion is d2s / dt2 + 16s = 0, where s is the stretch of the spring at time t. If s = 2 and ds/dt = 1 when t = 0, find s when t = 1/8

Accepted Answer

1.88

Answer

1.99

Answer

1.99

Answer

1.62

Question 37

A tank initially contains 400 gallons of brine in which 100 pounds of salt are dissolved. Pure water is running into the tank at the rate of 20 gallons per minute, and the maximum (which is kept uniform by stirring) is drained off at the same rate. How many pounds of salt remain in the tank after 30 minutes?

Accepted Answer

22.31

Answer

34.32

Answer

42.23

Answer

18.26

Question 38

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

Accepted Answer

2xdy – ydx = 0

Answer

ydx + ydx = 0

Answer

2ydx – xdy = 0

Answer

dy/dx – x = 0

Question 39

Find the orthogonal trajectories of the family of parabolas y^2 = 2x + c

Accepted Answer

y = Ce^-x

Answer

y = Ce^-2x

Answer

y = Ce^x

Answer

y = Ce^2x

Question 40

A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the substance remain after 9 years?

Accepted Answer

1.25 g

Answer

2.50 g

Answer

5.20 g

Answer

10.20 g

Question 41

Which of the following is an exact DE

Accepted Answer

2xydy + (2 + x^2) dy = 0

Answer

(x^2 + 1) dx – xydy = 0

Answer

xdy + (3x – 2y) dx = 0

Answer

x^2 ydy – ydx = 0

Question 42

A tank contains 1000 liters of a solution consisting 100 kg of salt dissolved in water. Pure water is pumped into the tank at the rate of 5 L/s, and the mixture kept uniform by stirring is pumped out at the same rate. How long will be required for 90 kg of salt to drain from the tank?

Accepted Answer

7 min 41 sec

Answer

6 min 58 sec

Answer

7 min 25 sec

Answer

8 min 02 sec

Question 43

A spherical tank of radius 4 ft is full of gasoline when a circular bottom hole with radius 1 in is opened. How long will be required for all the gasoline to drain from the tank?

Accepted Answer

14 min 29 sec

Answer

18 min 45 sec

Answer

12 min 38 sec

Answer

16 min 12 sec

Question 44

If dP / dQ = 2(sq.rt P) , find P

Accepted Answer

p = (Q sqr.) + 2QC + C^2

Answer

p = 2 (sqr.) + C

Answer

p = 4 (Q sqr.) + QC

Answer

p = (Q sqr.) + C

Question 45

A bacteria has a growth rate constant of 0.02. If initially the number of bacteria is 1000, what is the time before it reaches a population of 100,000?

Accepted Answer

230.26

Answer

205.87

Answer

209.22

Answer

208.36

Question 46

Solve the particular solution dx/dt = x/2 if x(0) = 1

Accepted Answer

x^2 = e^t

Answer

x^3 = e^t

Answer

x^4 = te^t

Answer

x = e^t

Question 47

The solution of the differential equation dy/dt = ky, with initial condition y = 1 when t = 0, is

Accepted Answer

y = e^kt

Answer

y^2 = 2t + k

Answer

y = sin kt

Answer

y = lnkt

Question 48

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

Accepted Answer

xdy – ydx = 0

Answer

ydx + xdy = 0

Answer

xdx + ydy = 0

Answer

xdx – ydy = 0

Question 49

What is the general solution of (D^4 – 1) y(t) = 0?

Accepted Answer

y = c₁e^t + c₂e^-t+ c₃ cost + c₄ sint

Answer

y = c₁e^t + c₂e^t + c₃e^-t + c₄t e^-t

Answer

y = c₁e^t+ c₂e^-t

Answer

y = c₁e^t+ c₂te^t

Question 50

A certain population of bacteria grows such that its rate of change is always proportional to the amount present. It doubles in 2 years. If in 3 years there are 20,000 of bacteria present, how much is present initially

Accepted Answer

7071

Answer

9071

Answer

8071

Answer

10071

Related quizzes

Related quizzes