() Write the given linear system in matrix form. Assume X = y dx = x - y + z +t - 1 dt dy = 4x + y – z – 8t2 dt dz = x + y + z + t2 - t + 4 dt -1 X' = X + t2 x + ? + t + 4
() Write the given linear system in matrix form. Assume X = y dx = x - y + z +t - 1 dt dy = 4x + y – z – 8t2 dt dz = x + y + z + t2 - t + 4 dt -1 X' = X + t2 x + ? + t + 4
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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![**Transcription:**
## Linear System in Matrix Form
Given the following system of differential equations, write it in matrix form. Assume \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).
\[
\frac{dx}{dt} = x - y + z + t - 1
\]
\[
\frac{dy}{dt} = 4x + y - z - 8t^2
\]
\[
\frac{dz}{dt} = x + y + z + t^2 - t + 4
\]
We need to express this system using matrices as follows:
\[
X' = \begin{pmatrix} \text{[ ]} & \text{[ ]} & \text{[ ]} \\ \text{[ ]} & \text{[ ]} & \text{[ ]} \\ \text{[ ]} & \text{[ ]} & \text{[ ]} \end{pmatrix} X + \begin{pmatrix} \text{[ ]} \\ \text{[ ]} \\ \text{[ ]} \end{pmatrix} t^2 + \begin{pmatrix} \text{[ ]} \\ \text{[ ]} \\ \text{[ ]} \end{pmatrix} t + \begin{pmatrix} -1 \\ 0 \\ 4 \end{pmatrix}
\]
**Explanation:**
The equations are represented in the matrix form \( X' = AX + B t^2 + Ct + D \), where:
- \( A \) is the coefficient matrix for \( X \).
- \( B \), \( C \), and \( D \) are vectors representing the coefficients for respective terms \( t^2 \), \( t \), and constant terms.
Fill in the respective values in the matrices to complete the system representation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20fa932-311a-46cd-86ab-78d74c804e39%2Fd6070541-6e67-4bf2-b320-e36b275d9687%2F2rzkewd_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription:**
## Linear System in Matrix Form
Given the following system of differential equations, write it in matrix form. Assume \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).
\[
\frac{dx}{dt} = x - y + z + t - 1
\]
\[
\frac{dy}{dt} = 4x + y - z - 8t^2
\]
\[
\frac{dz}{dt} = x + y + z + t^2 - t + 4
\]
We need to express this system using matrices as follows:
\[
X' = \begin{pmatrix} \text{[ ]} & \text{[ ]} & \text{[ ]} \\ \text{[ ]} & \text{[ ]} & \text{[ ]} \\ \text{[ ]} & \text{[ ]} & \text{[ ]} \end{pmatrix} X + \begin{pmatrix} \text{[ ]} \\ \text{[ ]} \\ \text{[ ]} \end{pmatrix} t^2 + \begin{pmatrix} \text{[ ]} \\ \text{[ ]} \\ \text{[ ]} \end{pmatrix} t + \begin{pmatrix} -1 \\ 0 \\ 4 \end{pmatrix}
\]
**Explanation:**
The equations are represented in the matrix form \( X' = AX + B t^2 + Ct + D \), where:
- \( A \) is the coefficient matrix for \( X \).
- \( B \), \( C \), and \( D \) are vectors representing the coefficients for respective terms \( t^2 \), \( t \), and constant terms.
Fill in the respective values in the matrices to complete the system representation.
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