Master Modeling with Differential Equations: Interactive Learning Quizzes

Question 1

A bacterial population grows exponentially at a rate proportional to its size. If the population doubles every hour, what is the differential equation that models this growth?

Accepted Answer

dP/dt = kP, where k is a positive constant.

Answer

dP/dt = kP², where k is a positive constant.

Answer

dP/dt = -kP

Answer

d²P/dt² = kP

Question 2

The concentration of a drug in the bloodstream decays exponentially with a half-life of 4 hours. If the initial concentration is 100 mg, what is the differential equation that models the concentration over time?

Accepted Answer

dC/dt = -kC, where k is a positive constant.

Answer

dC/dt = kC

Answer

d²C/dt² = -kC

Answer

C = ktdt

Question 3

A spring-mass system is described by the differential equation m(d²x/dt²) + kx = 0. What is the natural frequency of this system?

Accepted Answer

ω = √(k/m)

Answer

ω = k/m

Answer

ω = m/k

Answer

ω = √(m/k)

Question 4

The Van der Pol oscillator is described by the differential equation d²x/dt² - ε(1 - x²)dx/dt + x = 0.  What is the behavior of this oscillator when ε > 0?

Accepted Answer

The oscillator exhibits a limit cycle, which is a closed curve in the phase plane representing the long-term behavior of the system.

Answer

The oscillator approaches a stable equilibrium point.

Answer

The oscillator exhibits chaotic behavior.

Answer

The oscillator's amplitude grows unboundedly.

Question 5

Which of the following is NOT a numerical method for solving differential equations?

Accepted Answer

Gauss-Jordan elimination

Answer

Finite difference method

Answer

Runge-Kutta method

Answer

Euler's method

Question 6

What is the order of the following differential equation: d³x/dt³ + d²x/dt² + dx/dt + x = 0?

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 7

A population of yeast cells grows exponentially. The population doubles in size every hour. If the growth is modeled by the differential equation  *dy/dt = ky*, where *k* is a constant, what is the value of *k*?

Accepted Answer

k = ln(2)

Answer

k = 2

Answer

k = 1/2

Answer

k = 1/ln(2)

Question 8

A radioactive substance decays according to the differential equation *dQ/dt = -kQ*, where *k* is a constant. The substance has a half-life of 100 years. What is the value of *k*?

Accepted Answer

k = ln(2)/100

Answer

k = 2

Answer

k = -ln(2)/100

Answer

k = 1/100

Question 9

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

Accepted Answer

y' + 2xy = sin(x)

Answer

y'^3 + y' = x

Answer

y' = y^2

Answer

y'' + 2xy' = x^2

Question 10

Solve the differential equation *y' = x + y* with the initial condition *y(0) = 1*. What is the solution?

Accepted Answer

y = x + e^x

Answer

y = x + e^(-x)

Answer

y = x - e^x

Answer

y = x - e^(-x)

Question 11

Which of the following is a technique for solving separable differential equations?

Accepted Answer

Integrating both sides of the equation with respect to their respective variables.

Answer

Employing the Laplace transform.

Answer

Using the method of undetermined coefficients.

Answer

Applying the chain rule.

Question 12

Consider the system of differential equations: 
*dx/dt = x - y*
*dy/dt = x + y*. What is the critical point of this system?

Accepted Answer

(0, 0)

Answer

(1, 1)

Answer

(-1, -1)

Answer

(1, 0)

Question 13

Which of the following describes a type of stability for a critical point in a dynamical system?

Accepted Answer

Asymptotically stable

Answer

Saddle point

Answer

Unstable

Answer

Center

Question 14

What is the order of the differential equation *y^(4) + 2y'' + y = 0*?

Accepted Answer

4

Answer

8

Answer

2

Answer

6

Question 15

Which of the following represents a boundary condition for the differential equation *y'' + y = 0*?

Accepted Answer

y(0) = 0

Answer

y(0) + y(1) = 0

Answer

y'(0) = 1

Answer

y(1) = 0

Question 16

What is the Laplace transform of *f(t) = e^(-at)*, where *a* is a constant?

Accepted Answer

1/(s + a)

Answer

1/(s - a)

Answer

s/(s + a)

Answer

e^(-(s + a))