1.
What is the general solution of the following linear plane autonomous system of first order differential equations?
Correct Answer
D. Option 4
2.
If the general solution of a linear system of differential equation is the following. The eigenvalues of the coefficient matrix are:
Correct Answer(s)
A. Complex
B. Pure Imaginary
Explanation
The general solution of a linear system of differential equation can have complex eigenvalues, pure imaginary eigenvalues, or a combination of real and repeated eigenvalues. In this case, the correct answer indicates that the eigenvalues of the coefficient matrix can be both complex and pure imaginary.
3.
A dynamical system is ________ when the rate at which the system changes depends only on the present state of the system.
Correct Answer(s)
autonomous
Explanation
A dynamical system is considered autonomous when the rate at which the system changes depends only on the present state of the system. This means that the system evolves independently and does not rely on external factors or inputs to determine its behavior. The future behavior of the system is solely determined by its current state, making it autonomous and self-contained.
4.
The given system of first order differential equations is linear and autonomous?
Correct Answer
B. False
Explanation
A system of first order differential equations is said to be linear if the dependent variables and their derivatives appear only in linear terms. Autonomous means that the system does not explicitly depend on the independent variable. In this case, the question does not provide any information about the form of the system of equations, so it cannot be determined whether it is linear or autonomous. Therefore, the correct answer is False.
5.
For the following system of differential equations, the missing eigenvector is given by:
Correct Answer(s)
C. Option 3
D. Option 4
6.
(0,0) is the only critical point of a linear plane autonomous system of first order differential equations if for the coefficient matrix :
Correct Answer
B. Ad-bc ≠ 0
Explanation
The condition ad-bc ≠ 0 implies that the determinant of the coefficient matrix is non-zero. In a linear plane autonomous system, the critical points are the solutions where the derivative is zero. If ad-bc ≠ 0, it means that the coefficient matrix is invertible, and the system has a unique critical point at (0,0). This is because the determinant being non-zero ensures that the system of equations has a unique solution, which corresponds to the critical point.
7.
The non-zero critical point of the following system is/are:
Correct Answer(s)
B. (1,1)
C. (-1,-1)
Explanation
The non-zero critical points of the given system are (1,1) and (-1,-1). This means that when both x and y are equal to 1, or when both x and y are equal to -1, the system reaches a critical point where the derivative of both equations is zero. These points are significant because they represent the points where the system is at a turning point or a local extremum.
8.
The possible solutions for a plane autonomous system of first order differential equations are:
Correct Answer(s)
A. Cycle
B. An arc
D. Stationary point
Explanation
The possible solutions for a plane autonomous system of first order differential equations are a cycle, an arc, and a stationary point. A cycle refers to a closed curve where the system of equations repeats itself after a certain period of time. An arc refers to a portion of a curve that is not closed and does not cross itself. A stationary point refers to a point where the system of equations remains constant and does not change over time.
9.
Select the phase portrait of a linear plane autonomous system of differential equations if the eigenvalues of the coefficient matrix are real with opposite signs.
Correct Answer
A. Option 1
Explanation
Option 1 is the correct answer because when the eigenvalues of the coefficient matrix are real with opposite signs, it indicates a stable saddle point in the phase portrait. In Option 1, the phase portrait shows a saddle point, which is consistent with the given condition.