4- (a) X' = In the following problems solve the given initial value problem. HIN 2 — X X(0) = (²³)
4- (a) X' = In the following problems solve the given initial value problem. HIN 2 — X X(0) = (²³)
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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Question
![**Problem 4: Initial Value Problems**
In the following problems, solve the given initial value problem.
**(a)**
\[
X' =
\begin{pmatrix}
\frac{1}{2} & 0 \\
1 & -\frac{1}{2}
\end{pmatrix}
X, \quad
X(0) =
\begin{pmatrix}
3 \\
5
\end{pmatrix}
\]
**(b)**
\[
X' =
\begin{pmatrix}
1 & 1 & 4 \\
0 & 2 & 0 \\
1 & 1 & 1
\end{pmatrix}
X, \quad
X(0) =
\begin{pmatrix}
1 \\
3 \\
0
\end{pmatrix}
\]
**Explanation:**
- Part (a) presents a 2x2 matrix differential equation with an initial vector condition.
- Part (b) presents a 3x3 matrix differential equation with an initial vector condition.
To solve these, consider finding the eigenvalues and eigenvectors of the matrices to form the general solution, and then apply the initial condition to find the specific solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0a46b8d2-68f8-4ff5-85d9-0145753bf32f%2F9dc28f60-6c27-4f8a-9495-b35586feff1e%2Ft5758se_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Initial Value Problems**
In the following problems, solve the given initial value problem.
**(a)**
\[
X' =
\begin{pmatrix}
\frac{1}{2} & 0 \\
1 & -\frac{1}{2}
\end{pmatrix}
X, \quad
X(0) =
\begin{pmatrix}
3 \\
5
\end{pmatrix}
\]
**(b)**
\[
X' =
\begin{pmatrix}
1 & 1 & 4 \\
0 & 2 & 0 \\
1 & 1 & 1
\end{pmatrix}
X, \quad
X(0) =
\begin{pmatrix}
1 \\
3 \\
0
\end{pmatrix}
\]
**Explanation:**
- Part (a) presents a 2x2 matrix differential equation with an initial vector condition.
- Part (b) presents a 3x3 matrix differential equation with an initial vector condition.
To solve these, consider finding the eigenvalues and eigenvectors of the matrices to form the general solution, and then apply the initial condition to find the specific solution.
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