Solve the given initial-value problem. -(-) 8 -1 -9- X' = 5 x, X(0) = %3D 7t X(t) = cos(: 8 7t -7 sin(: + -19
Solve the given initial-value problem. -(-) 8 -1 -9- X' = 5 x, X(0) = %3D 7t X(t) = cos(: 8 7t -7 sin(: + -19
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please writer answer as that form -2C_1e^{14t}-C_2e^{-t}
![**Solve the given initial-value problem.**
Given the matrix differential equation:
\[
\mathbf{X'} =
\begin{pmatrix}
8 & -1 \\
5 & 6
\end{pmatrix}
\mathbf{X}, \quad \mathbf{X}(0) =
\begin{pmatrix}
-6 \\
8
\end{pmatrix}
\]
The solution is expressed as:
\[
\mathbf{X}(t) =
e^{7t \begin{pmatrix} \frac{-6}{8} \end{pmatrix}} \cos(2t)
+ e^{7t \begin{pmatrix} \frac{-7}{-19} \end{pmatrix}} \sin(2t)
\]
**Explanation of Elements:**
- \(\mathbf{X'}\) represents the derivative of the matrix \(\mathbf{X}\) with respect to time \(t\).
- The given matrix inside the derivative equation indicates the system of differential equations.
- The initial condition \(\mathbf{X}(0)\) specifies the state of the system at time \(t=0\).
- The solution \(\mathbf{X}(t)\) involves exponential terms and trigonometric functions \( \cos(2t) \) and \( \sin(2t) \).
**Note:** There seems to be an error marked by a red cross next to the solution indicating a possible mistake in the expression or calculation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20fa932-311a-46cd-86ab-78d74c804e39%2F5dc3031f-6850-4dba-bbf0-cc1c0d1f57b0%2Flhecx24_processed.png&w=3840&q=75)
Transcribed Image Text:**Solve the given initial-value problem.**
Given the matrix differential equation:
\[
\mathbf{X'} =
\begin{pmatrix}
8 & -1 \\
5 & 6
\end{pmatrix}
\mathbf{X}, \quad \mathbf{X}(0) =
\begin{pmatrix}
-6 \\
8
\end{pmatrix}
\]
The solution is expressed as:
\[
\mathbf{X}(t) =
e^{7t \begin{pmatrix} \frac{-6}{8} \end{pmatrix}} \cos(2t)
+ e^{7t \begin{pmatrix} \frac{-7}{-19} \end{pmatrix}} \sin(2t)
\]
**Explanation of Elements:**
- \(\mathbf{X'}\) represents the derivative of the matrix \(\mathbf{X}\) with respect to time \(t\).
- The given matrix inside the derivative equation indicates the system of differential equations.
- The initial condition \(\mathbf{X}(0)\) specifies the state of the system at time \(t=0\).
- The solution \(\mathbf{X}(t)\) involves exponential terms and trigonometric functions \( \cos(2t) \) and \( \sin(2t) \).
**Note:** There seems to be an error marked by a red cross next to the solution indicating a possible mistake in the expression or calculation.
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