Consider the following initial-value problem. 2 x' = X, X(0) = 1 Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenval K, K, = Solve the given initial-value problem. X(t) =

Consider the following initial-value problem. 2 x' = X, X(0) = 1 Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenval K, K, = Solve the given initial-value problem. X(t) =

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

### Initial-Value Problem

Consider the following initial-value problem:

\[ \mathbf{X}' =
\begin{pmatrix}
\frac{1}{2} & 0 \\
1 & -\frac{1}{2}
\end{pmatrix}
\mathbf{X}, \quad \mathbf{X}(0) =
\begin{pmatrix}
2 \\
8
\end{pmatrix} \]

#### Tasks:

1. **Find the eigenvalues of the coefficient matrix $ A(t) $**. (Enter your answers as a comma-separated list.)

\[
\lambda = \_\_\_\_
\]

2. **Find an eigenvector for the corresponding eigenvalues**. (Enter your answers from smallest eigenvalue to largest eigenvalue.)

\[
\mathbf{K_1} =
\begin{pmatrix}
\_\_\_ \\
\_\_\_
\end{pmatrix}
\]

\[
\mathbf{K_2} =
\begin{pmatrix}
\_\_\_ \\
\_\_\_
\end{pmatrix}
\]

3. **Solve the given initial-value problem**.

\[
\mathbf{X}(t) = \_\_\_\_
\]

This problem involves finding eigenvalues and eigenvectors of the given coefficient matrix, and then using them to solve the initial-value problem.

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