1- a) b) C) d) e) sin t sin t 0 Let x₁(t) [] sin sin t sin t 0 sint 1. The Wronskian of ₁ (t), x2 (t), x3 (t) is W (x1, x2, x3)(t) = 2 sin³ t. II. x₁ (t), x₂ (t), x3 (t) are linearly independent on (-∞, ∞). III. x₁ (t), x₂ (t), x3 (t) are solutions of a linear homogeneous system x' (t) = A(t)x(t) on (-∞, ∞), where A(t) is a 3 x 3 matrix function continuous on (-∞, ∞). II and III I and III I and II I, II and III Only I Boş bırak , x₂ (t) = , x3 (t) Which of the following statements are true?
1- a) b) C) d) e) sin t sin t 0 Let x₁(t) [] sin sin t sin t 0 sint 1. The Wronskian of ₁ (t), x2 (t), x3 (t) is W (x1, x2, x3)(t) = 2 sin³ t. II. x₁ (t), x₂ (t), x3 (t) are linearly independent on (-∞, ∞). III. x₁ (t), x₂ (t), x3 (t) are solutions of a linear homogeneous system x' (t) = A(t)x(t) on (-∞, ∞), where A(t) is a 3 x 3 matrix function continuous on (-∞, ∞). II and III I and III I and II I, II and III Only I Boş bırak , x₂ (t) = , x3 (t) Which of the following statements are true?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Understanding Systems of Differential Equations
Consider the functions \( x_1(t) \), \( x_2(t) \), and \( x_3(t) \) given by:
\[ x_1(t) = \begin{bmatrix} \sin t \\ 0 \\ \sin t \end{bmatrix}, \quad x_2(t) = \begin{bmatrix} \sin t \\ \sin t \\ 0 \end{bmatrix}, \quad x_3(t) = \begin{bmatrix} 0 \\ \sin t \\ \sin t \end{bmatrix}. \]
We want to determine which of the following statements regarding these functions are true:
1. The Wronskian of \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) is \( W(x_1, x_2, x_3)(t) = 2 \sin^3 t \).
2. \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) are linearly independent on \((-\infty, \infty)\).
3. \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) are solutions of a linear homogeneous system \( x'(t) = A(t)x(t) \) on \((-\infty, \infty)\), where \( A(t) \) is a \( 3 \times 3 \) matrix function continuous on \((-\infty, \infty)\).
### Answer Choices
a) II and III
b) I and III
c) I and II
d) I, II and III
e) Only I
#### Explanation of Graphs/Diagrams
There are no graphs or diagrams in this problem. Instead, the problem focuses on vector functions and their properties.
- **Wronskian**: The Wronskian is a determinant used in the study of differential equations to determine whether a set of solutions is linearly independent.
- **Linear Independence**: A set of functions is linearly independent if no function in the set can be written as a linear combination of the others.
- **Linear Homogeneous System**: The differential equation \( x'(t) = A(t)x(t) \) represents a system where the rate of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7eb16600-8f07-40f7-854e-83024f28a747%2Fb8364f8d-7d0c-46f7-9cad-5d1ecbc7e08e%2Frquyce_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Systems of Differential Equations
Consider the functions \( x_1(t) \), \( x_2(t) \), and \( x_3(t) \) given by:
\[ x_1(t) = \begin{bmatrix} \sin t \\ 0 \\ \sin t \end{bmatrix}, \quad x_2(t) = \begin{bmatrix} \sin t \\ \sin t \\ 0 \end{bmatrix}, \quad x_3(t) = \begin{bmatrix} 0 \\ \sin t \\ \sin t \end{bmatrix}. \]
We want to determine which of the following statements regarding these functions are true:
1. The Wronskian of \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) is \( W(x_1, x_2, x_3)(t) = 2 \sin^3 t \).
2. \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) are linearly independent on \((-\infty, \infty)\).
3. \( x_1(t) \), \( x_2(t) \), \( x_3(t) \) are solutions of a linear homogeneous system \( x'(t) = A(t)x(t) \) on \((-\infty, \infty)\), where \( A(t) \) is a \( 3 \times 3 \) matrix function continuous on \((-\infty, \infty)\).
### Answer Choices
a) II and III
b) I and III
c) I and II
d) I, II and III
e) Only I
#### Explanation of Graphs/Diagrams
There are no graphs or diagrams in this problem. Instead, the problem focuses on vector functions and their properties.
- **Wronskian**: The Wronskian is a determinant used in the study of differential equations to determine whether a set of solutions is linearly independent.
- **Linear Independence**: A set of functions is linearly independent if no function in the set can be written as a linear combination of the others.
- **Linear Homogeneous System**: The differential equation \( x'(t) = A(t)x(t) \) represents a system where the rate of
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