Systems of Differential Equations: Master the Concepts for Complex Systems

Question 1

An eigenvalue is a scalar that, when multiplied by an associated vector, produces:

Accepted Answer

the same vector

Answer

a different vector

Answer

a vector with a different magnitude

Answer

a vector with a different direction

Question 2

To solve a system of differential equations with constant coefficients, one must determine its:

Accepted Answer

eigenvalues and eigenvectors

Answer

phase portrait

Answer

general solution

Answer

equilibrium points

Question 3

In a system of differential equations, a zero eigenvalue indicates that the system has a:

Accepted Answer

constant solution

Answer

unstable solution

Answer

periodic solution

Answer

no particular significance

Question 4

Consider the system of differential equations: dx/dt = x + 2y, dy/dt = 3x - 4y. The eigenvalues of this system are:

Accepted Answer

-1, -4

Answer

-1, 4

Answer

1, -4

Answer

1, 4

Question 5

To diagonalize a matrix, one can use a:

Accepted Answer

similarity transformation

Answer

orthogonal transformation

Answer

rotation transformation

Answer

permutation transformation

Question 6

In a phase portrait of a system of differential equations, a stable node represents a point where trajectories:

Accepted Answer

approach the point

Answer

diverge from the point

Answer

intersect at the point

Answer

oscillate around the point

Question 7

Find the solution to the system of differential equations: dx/dt = y - 1, dy/dt = -x + 2

Accepted Answer

x = 1 + Ce^(-t), y = 2 + Ce^(-t)

Answer

x = 1 - Ce^(-t), y = 2 - Ce^(-t)

Answer

x = e^(t) + Ce^(-t), y = 2e^(t) + Ce^(-t)

Answer

x = 1 + e^(-t), y = 2 + e^(-t)

Question 8

Find a fundamental matrix for the system dx/dt = y, dy/dt = -x + 2y:

Accepted Answer

[e^(t), e^(-t)][1, 1]

Answer

[e^(2t), e^(-t)][1, 1]

Answer

[e^(t), e^(2t)][1, 1]

Answer

[e^(2t), e^(-2t)][1, 1]

Question 9

A radioactive substance decays at a rate proportional to the amount present. Given that the half-life of the substance is 10 years, find the decay constant:

Accepted Answer

0.1386

Answer

0.0693

Answer

0.2773

Answer

0.5546

Question 10

Find a Lyapunov function for the system dx/dt = 2x - y, dy/dt = x + y:

Accepted Answer

x^2 + y^2

Answer

x^2 - y^2

Answer

y^2 - x^2

Answer

-x^2 - y^2

Question 11

Which condition must be satisfied for a system of differential equations to have a periodic solution?

Accepted Answer

The system must be autonomous

Answer

The system must be linear

Answer

The system must have a constant Jacobian

Answer

The system must be Hamiltonian

Question 12

In the context of systems of differential equations, which of the following is NOT a recognized type of stability?

Accepted Answer

Periodically stable

Answer

Asymptotically stable

Answer

Unstable

Answer

Globally stable

Question 13

Which of the following is a real eigenvalue of the matrix $A = \begin{bmatrix} 2 & -3 \ 1 & 4 \end{bmatrix} $?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 14

Which of the following is true about the eigenvalues of a system of differential equations?

Accepted Answer

They are complex and conjugate.

Answer

They are real and distinct.

Answer

They are real and equal.

Answer

They can be any real numbers.

Question 15

Which of the following is the most appropriate characteristic equation for a system of differential equations?

Accepted Answer

det(A - λI) = 0

Answer

det(A + λI) = 0

Answer

λ² = 0

Answer

λ² + 1 = 0

Question 16

Consider the system of differential equations: dx/dt = -2x + y, dy/dt = x - 3y. Find the general solution to the system.

Accepted Answer

x(t) = e^(-t)cos(2t) + 2e^(-t)sin(2t), y(t) = e^(-t)cos(2t) - e^(-t)sin(2t)

Answer

x(t) = e^(-t)cos(2t) - 2e^(-t)sin(2t), y(t) = e^(-t)cos(2t) + e^(-t)sin(2t)

Answer

x(t) = e^tcos(2t) + 2e^tsin(2t), y(t) = e^tcos(2t) - e^tsin(2t)

Answer

x(t) = e^tcos(2t) - 2e^tsin(2t), y(t) = e^tcos(2t) + e^tsin(2t)

Question 17

In a population growth model, what does the carrying capacity represent?

Accepted Answer

The maximum population size that can be sustained by the environment

Answer

The rate at which the population grows

Answer

The number of individuals that die each year

Answer

The initial population size

Question 18

Consider the system of differential equations dx/dt = x(y - 1), dy/dt = y(x - 1). What is the critical point for this system?

Accepted Answer

(1, 1)

Answer

(0, 0)

Answer

(0, 1)

Answer

(1, 0)

Question 19

Which of the following is a qualitative analysis technique that can be used to study the behavior of a system of differential equations without solving it explicitly?

Accepted Answer

Phase plane analysis

Answer

Numerical simulation

Answer

Perturbation theory

Answer

Lyapunov stability analysis

Question 20

Consider the system of differential equations dx/dt = ax + by, dy/dt = cx + dy. Identify the eigenvalue of the coefficient matrix [[a, b], [c, d]]

Accepted Answer

a + d

Answer

a - d

Answer

c - b

Answer

c + b

Question 21

Determine the eigenvector corresponding to the eigenvalue λ = 2 of the matrix [[3, 2], [-1, 1]]

Accepted Answer

[-1, 1]

Answer

[1, -1]

Answer

[2, 1]

Answer

[-2, 1]

Question 22

Consider the system of differential equations: dx/dt = 2x + y, dy/dt = -x + 2y. Which of the following is an eigenvalue of the coefficient matrix?

Accepted Answer

-1

Answer

1

Answer

2

Answer

3

Question 23

Given the system of differential equations: dx/dt = x - 2y, dy/dt = 2x + y. What is the corresponding characteristic equation?

Accepted Answer

λ² - λ - 2 = 0

Answer

λ² + λ - 2 = 0

Answer

λ² - 2λ + 1 = 0

Question 24

Consider the system of differential equations: dx/dt = ax + by, dy/dt = cx + dy. If the eigenvalues are complex, what can be concluded about the behavior of the solutions?

Accepted Answer

The solutions will exhibit oscillatory behavior.

Answer

The solutions will converge to the origin.

Answer

The solutions will be constant.

Answer

The solutions will diverge from the origin.

Question 25

In a population dynamics model described by the system of differential equations: dx/dt = a - bx, dy/dt = cx - dy. What does the term "a - bx" represent?

Accepted Answer

The growth rate of species x in the absence of competition.

Answer

The rate of competition between species x and y.

Answer

The carrying capacity for species x.

Question 26

Which of the following is a necessary condition for a system of differential equations to have a unique solution?

Accepted Answer

The coefficient matrix must be non-singular.

Answer

The initial conditions must be unique.

Answer

The system must be linear.

Question 27

Consider the system of differential equations: dx/dt = y, dy/dt = x. What is the nature of the equilibrium point at (0, 0)?

Accepted Answer

Center

Answer

Stable node

Answer

Unstable node

Answer

Saddle point

Question 28

Which of the following methods can be used to solve a system of differential equations with constant coefficients?

Accepted Answer

Eigenvalue decomposition

Answer

Separation of variables

Answer

Integrating factor

Answer

Variation of parameters

Question 29

Consider the system of differential equations: dx/dt = x + 2y, dy/dt = -x + y. What is the trace of the coefficient matrix?

Accepted Answer

3

Answer

2

Answer

-1

Answer

1

Question 30

Which of the following is NOT a property of an eigenvector?

Accepted Answer

It is orthogonal to all other eigenvectors.

Answer

It is a non-zero vector.

Answer

It scales the solution.

Answer

It corresponds to an eigenvalue.

Question 31

Consider the system of differential equations: dx/dt = x - y, dy/dt = x + y. What type of equilibrium point is (0, 0)?

Accepted Answer

Saddle point

Answer

Stable node

Answer

Unstable node

Answer

Center