Applications of First-Order Differential Equations: Interactive Learning Quizzes

Question 1

Identify the initial value problem for the differential equation y' = sin(x) that has a solution satisfying y(0) = 1.

Accepted Answer

y' = sin(x), y(0) = 1

Answer

y' = sin(x), y(0) = 0

Answer

y' = sin(x) + 1, y(0) = 0

Answer

y' = sin(x) - 1, y(0) = 1

Question 2

Formulate the differential equation that models the charge q(t) in a circuit with inductance L, resistance R, and capacitance C.

Accepted Answer

Lq'' + Rq' + 1/C * q = 0

Answer

Lq'' + Rq' - 1/C * q = 0

Answer

Rq'' + Lq' + 1/C * q = 0

Answer

Rq'' + Lq' - 1/C * q = 0

Question 3

A population of rabbits grows at a rate proportional to its current population. If the initial population is 100 and doubles in 5 years, what will be the population after 10 years?

Accepted Answer

1000

Answer

500

Answer

750

Answer

250

Question 4

Which of the following is NOT a common method for solving first-order differential equations?

Accepted Answer

Runge-Kutta method

Answer

Separation of variables

Answer

Integrating factors

Answer

Substitution

Question 5

Consider the differential equation y' + 2xy = x. What type of first-order differential equation is this?

Accepted Answer

Linear

Answer

Bernoulli

Answer

Riccati

Answer

Exact

Question 6

The differential equation dy/dx = (x + y) / (x - y) has a singular point at:

Accepted Answer

(x, y) = (0, 0)

Answer

x = 0

Answer

y = 0

Answer

x = y

Question 7

Which of the following is a necessary condition for a first-order differential equation to be exact?

Accepted Answer

The partial derivative of P with respect to y is equal to the partial derivative of Q with respect to x

Answer

The partial derivative of P with respect to x is equal to the partial derivative of Q with respect to y

Answer

The function P(x, y) is independent of y

Answer

The function Q(x, y) is independent of x

Question 8

Consider the differential equation y' = f(x)y + g(x). What is the integrating factor for this equation?

Accepted Answer

e^(∫f(x)dx)

Answer

e^(-∫f(x)dx)

Answer

1 / f(x)

Answer

1 / g(x)

Question 9

Which of the following is a qualitative analysis technique for first-order differential equations?

Accepted Answer

Slope fields

Answer

Euler's method

Answer

Separation of variables

Answer

Laplace transforms

Question 10

A population of bacteria grows exponentially with a growth rate proportional to the population size. If the population doubles in size in 5 hours, which differential equation models this growth?

Accepted Answer

dy/dt = ky, where k is a positive constant

Answer

dy/dt = -ky, where k is a positive constant

Answer

dy/dt = k, where k is a constant

Answer

dy/dt = -ky, where k is a negative constant

Question 11

Which of the following equations is a first-order linear differential equation?

Accepted Answer

y' - 2xy = x

Answer

y' + y^2 = 0

Answer

y'' + y' + y = 0

Answer

xy' + y = e^x

Question 12

Find the general solution to the differential equation dy/dx = x/(y + 1).

Accepted Answer

ln(y+1) = (x^2)/2 + C

Answer

y = (x^2)/2 + C

Answer

y = x^2 + C

Answer

ln(y+1) = x^2 + C

Question 13

What is the order of the differential equation d^4y/dx^4 - 3y'' + 2y = sin(x)?

Accepted Answer

Fourth order

Answer

First order

Answer

Second order

Answer

Third order

Question 14

Which of the following methods can be used to solve exact differential equations?

Accepted Answer

Multiplying both sides by an integrating factor

Answer

Integrating both sides with respect to x

Answer

Substituting y' into the original equation

Answer

Using separation of variables

Question 15

The differential equation dP/dt = P(1 - P) models a population. Without solving the equation, what best describes the equilibrium of the population at P = 0?

Accepted Answer

Unstable equilibrium

Answer

Stable equilibrium

Answer

Unstable equilibrium at P = 1

Answer

Stable equilibrium at P = 1

Question 16

Find the equilibrium solutions for the differential equation: dy/dx = y(1-y)

Accepted Answer

y = 0

Accepted Answer

y = 1

Answer

y = -1

Answer

y = 2

Question 17

The velocity of a particle moving along a straight line is given by v(t) = 2t^2 + 3t - 5 m/s. If the particle starts at the origin, find its displacement after 5 seconds.

Accepted Answer

35 meters

Answer

25 meters

Answer

55 meters

Answer

45 meters

Question 18

A cup of coffee cools down according to Newton's law of cooling. The temperature of the coffee is initially 95 degrees Celsius and cools to 80 degrees Celsius in 5 minutes when the surrounding temperature is 20 degrees Celsius. What is the temperature of the coffee after **10 minutes**?

Accepted Answer

62.4

Answer

70.0

Answer

47.6

Answer

55.0

Question 19

A radioactive substance decays exponentially with a half-life of 10 days. If the initial amount of the substance is 200 mg, what is the amount remaining after **25 days**?

Accepted Answer

50

Answer

100

Answer

75

Question 20

The velocity of a falling object is given by the differential equation dv/dt = g - kv, where g is the acceleration due to gravity and k is a constant. If the object starts at rest, what is the velocity of the object after **t seconds**?

Accepted Answer

v = (g/k)(1 - e^(-kt))

Answer

v = gte^(-kt)

Answer

v = gt - (g/k)

Answer

v = (g/k)e^(-kt)

Question 21

A chemical reaction follows the first-order rate law: dC/dt = -kC, where C is the concentration of the reactant and k is the rate constant. If the initial concentration is 1 mol/L and the rate constant is 0.05 min^(-1), what is the time it takes for the concentration to reach 0.25 mol/L?

Accepted Answer

13.86

Answer

10.00

Answer

15.23

Answer

11.51

Question 22

A population of animals is growing according to the logistic equation: dP/dt = 0.5P(1 - P/1000), where P is the population size. If the initial population is 100, what is the carrying capacity of the environment?

Accepted Answer

1000

Answer

2000

Answer

1500

Answer

500

Question 23

A radioactive substance with an initial amount of 100 grams decays exponentially with a decay constant of 0.05 per year. How many grams of the substance remain after 10 years?

Accepted Answer

37.25

Answer

10.5

Answer

62.75

Question 24

A car traveling at 60 miles per hour applies the brakes and decelerates at a rate of 10 feet per second squared. How far (in feet) will the car travel before coming to a stop?

Accepted Answer

2700

Answer

1350

Answer

4500

Answer

3600

Question 25

An electric circuit has a resistance of 10 ohms, an inductance of 0.1 henrys, and a capacitance of 0.01 farads. If the charge on the capacitor is initially 0 and the current is 1 ampere, what is the charge on the capacitor after 0.1 seconds?

Accepted Answer

Approximately 0.0099 coulombs

Answer

0.001 coulombs

Answer

0.00099 coulombs

Answer

0.01 coulombs

Question 26

A mechanical system is described by the differential equation d2x/dt2 + 4dx/dt + 4x = 0. What is the characteristic equation of the system?

Accepted Answer

r^2 + 4r + 4 = 0

Answer

r^2 + 4r - 4 = 0

Answer

r^2 - 4r - 4 = 0

Answer

r^2 - 4r + 4 = 0

Question 27

A population of fish is introduced into a lake and their growth is modeled by the differential equation dP/dt = 0.5P(1 - P/500), where P is the population size. If the initial population is 100 fish, what is the carrying capacity of the lake?

Accepted Answer

500 fish

Answer

250 fish

Answer

1000 fish

Answer

750 fish

Question 28

A tank initially contains 100 gallons of water with a salt concentration of 0.1 lb/gal. Salt water with a concentration of 0.2 lb/gal flows into the tank at a rate of 5 gal/min, and the well-mixed solution flows out at the same rate. What is the salt concentration in the tank after 10 minutes?

Accepted Answer

0.15 lb/gal

Answer

0.2 lb/gal

Answer

0.1 lb/gal

Question 29

A predator-prey model describes the population dynamics of a predator and a prey species. The rate of change of the predator population is proportional to the product of the predator and prey populations. Which of the following differential equations represents this model?

Accepted Answer

dP/dt = kPQ

Answer

dP/dt = k(P/Q)

Answer

dP/dt = k(P-Q)

Answer

dP/dt = kP + kQ

Question 30

A spring-mass system is governed by the differential equation m(d^2x/dt^2) + kx = 0. What is the general solution to this equation?

Accepted Answer

x = Acos(sqrt(k/m)t) + Bsin(sqrt(k/m)t)

Answer

x = Ae^(t) + Be^(-t)

Answer

x = Ae^(sqrt(k/m)t) + Be^(-sqrt(k/m)t)

Answer

x = Acos(t) + Bsin(t)

Question 31

A differential equation that describes the population of a species with limited resources is given by dN/dt = rN(1 - N/K), where r is the intrinsic growth rate and K is the carrying capacity. What is the equilibrium population size?

Accepted Answer

K

Answer

K/2

Answer

r

Answer

0

Question 32

A tank initially contains 100 liters of pure water. A brine solution containing 0.5 kg of salt per liter is pumped into the tank at a rate of 5 liters per minute. The well-mixed solution is pumped out at the same rate. What is the amount of salt in the tank after 10 minutes?

Accepted Answer

23.86 kg

Answer

25 kg

Answer

50 kg

Answer

12.5 kg

Question 33

What is the general solution to the differential equation dy/dx = y/x?

Accepted Answer

y = Cx

Answer

y = x^2 + C

Answer

y = Ce^x

Answer

y = x + C

Question 34

A radioactive substance decays at a rate proportional to its mass. Its half-life is 10 years. How long will it take for 90% of the substance to decay?

Accepted Answer

33.22 years

Answer

10 years

Answer

50 years

Answer

20 years

Question 35

What is the solution to the initial value problem dy/dx = y^2, with the initial condition y(0) = 1?

Accepted Answer

y = 1/(1-x)

Answer

y = 1/(1+x)

Answer

y = e^x

Answer

y = x^2 + 1

Question 36

A tank contains 100 gallons of water with 5 pounds of salt dissolved. Brine containing 0.2 pounds of salt per gallon enters the tank at 3 gallons/minute. The well-mixed solution exits at the same rate. What is the limiting amount of salt in the tank as time approaches infinity?

Accepted Answer

20 pounds

Answer

0 pounds

Answer

10 pounds

Answer

5 pounds

Question 37

What is the general solution of the differential equation dy/dx = x^2y?

Accepted Answer

y = Ce^(x^3/3)

Answer

y = Ce^(x/3)

Answer

y = Ce^(x^2/2)

Answer

y = Ce^(x^3)

Question 38

What is the general solution of the differential equation y' + 2xy = x?

Accepted Answer

y = (1/2) + Ce^(-x^2)

Answer

y = (1/2)x^2 + Ce^(x^2)

Answer

y = Ce^(-x^2)

Answer

y = x^2 + Ce^(-x)

Question 39

Which of the following is NOT a technique commonly used to solve first-order differential equations?

Accepted Answer

Laplace Transforms

Answer

Bernoulli equations

Answer

Integrating factors

Answer

Exact equations

Question 40

What is the order and degree of the differential equation  d²y/dx² - 3 dy/dx + 2y = cos(x)?

Accepted Answer

Order: 2, Degree: 1

Answer

Order: 2, Degree: 2

Answer

Order: 1, Degree: 2

Answer

Order: 1, Degree: 1

Question 41

Which of the following represents the initial condition for the differential equation dy/dx = x + y, given that y = 1 when x = 0?

Accepted Answer

y(0) = 1

Answer

y(1) = 0

Answer

x(0) = 1

Answer

y'(0) = 1

Question 42

Solve the differential equation dy/dx = y/x, with the initial condition y(1) = 2.

Accepted Answer

y = 2x

Answer

y = 2/x

Answer

y = x/2

Answer

y = -2x

Question 43

Which of the following is a common application of first-order differential equations in the field of biology?

Accepted Answer

Modeling population growth

Answer

Calculating the speed of light

Answer

Predicting the weather

Answer

Designing bridges

Question 44

A parachutist weighing 160 lb falls from an airplane. The air resistance they experience is proportional to the square of their velocity. Given an initial velocity of 0 ft/s and a terminal velocity of 120 ft/s, what is the parachutist's velocity after 5 seconds of freefall?

Accepted Answer

82.17 ft/s (approximately)

Answer

108.51 ft/s

Answer

95.34 ft/s

Answer

69.00 ft/s

Question 45

The current in an electrical circuit is described by the function i(t) = 10e^(-0.5t), where t is time in seconds. At what time does the current reach its maximum value?

Accepted Answer

0 seconds

Answer

3 seconds

Answer

1 second

Answer

2 seconds

Question 46

A rectangular garden's length is 3 meters longer than its width. If the perimeter is increasing at a rate of 1 meter per year, what is the rate of change of the width when the width is 10 meters?

Accepted Answer

1/13 meter/year

Answer

1/15 meter/year

Answer

1/17 meter/year

Answer

1/11 meter/year

Question 47

A tank initially holds 100 liters of pure water. Brine containing 2 kg of salt per liter is pumped into the tank at a rate of 5 liters per minute. The well-mixed solution is then pumped out at the same rate. How much salt is in the tank after 10 minutes?

Accepted Answer

76.84 kg (approximately)

Answer

100 kg

Answer

25 kg

Answer

50 kg

Question 48

What is the general solution to the differential equation dy/dx = y/x?

Accepted Answer

y = Cx

Answer

y = Cx^2

Answer

y = x + C

Answer

y = Ce^x

Question 49

The differential equation dy/dx = y² - 1 has a critical point at y = 1. What is the stability of this critical point?

Accepted Answer

Unstable

Answer

Semi-stable

Answer

Stable

Answer

Asymptotically stable

Question 50

A circuit consists of a resistor, a capacitor, and a battery. The voltage across the capacitor, V(t), satisfies the differential equation dV/dt = (1/RC)(V_b - V), where V_b is the battery voltage, R is the resistance, and C is the capacitance. What is the time constant of this circuit?

Accepted Answer

RC

Answer

1/RC

Answer

C/R

Answer

R/C

Question 51

The solution to the initial value problem dy/dx = 2x, y(0) = 1 is:

Accepted Answer

y = x^2 + 1

Answer

y = x + 1

Answer

y = x^2 - 1

Answer

y = 2x + 1