---### Linear Algebra: General Solutions to Differential EquationsIn this exercise, we aim to find the general solution to the differential equation given by $ x' = Ax $ where $ A $ is the provided matrix.---#### Problem Statement:For each problem, find the general solution to $ x' = Ax $, where $ A $ is the given matrix.---#### (a) \[ A = \begin{pmatrix} 2 & 0 \\ -4 & -2 \end{pmatrix} \]---#### (b)\[ A = \begin{pmatrix}-4 & -8 \\4 & 4\end{pmatrix} \]---### Explanation:To determine the general solution for the differential equation $ x' = Ax $:1. **Find the Eigenvalues**: Solve $\det(A - \lambda I) = 0$ to find the eigenvalues $\lambda$. 2. **Find the Eigenvectors**: For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = 0$ to find the eigenvectors $\mathbf{v}$.3. **Construct the General Solution**: General solution will be of the form $x(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2$, where $\lambda_1, \lambda_2$ and $\mathbf{v}_1, \mathbf{v}_2$ are the eigenvalues and corresponding eigenvectors, and $c_1, c_2$ are arbitrary constants.---Utilize this procedure to find the solutions for the matrices provided in problems (a) and (b). If additional support or graphical descriptions are necessary, consult supplementary learning materials or a tutor. Here is the transcription of the text from the provided image for educational purposes:---**(c) $ \mathbf{A} = \begin{bmatrix} -2 & -1 \\ 4 & 2 \end{bmatrix} $.**---In the above equation, matrix $ \mathbf{A} $ is given as a 2x2 matrix with the elements arranged in two rows and two columns. The top-left element of the matrix is -2, top-right is -1, bottom-left is 4, and bottom-right is 2.Such matrices are commonly used in linear algebra to perform various operations such as matrix multiplication, finding determinants, and solving systems of linear equations.

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Answered: 3. For each problem, find the general…

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3. For each problem, find the general solution to x' = Ax, where A is the given matrix. (a) A = -4 (b) A =

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

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### Linear Algebra: General Solutions to Differential Equations

In this exercise, we aim to find the general solution to the differential equation given by $ x' = Ax $ where $ A $ is the provided matrix.

---

#### Problem Statement:

For each problem, find the general solution to $ x' = Ax $, where $ A $ is the given matrix.

---

#### (a)
\[ A = \begin{pmatrix} 2 & 0 \\ -4 & -2 \end{pmatrix} \]

---

#### (b)
\[ A = \begin{pmatrix}
-4 & -8 \\
4 & 4
\end{pmatrix} \]

---

### Explanation:

To determine the general solution for the differential equation $ x' = Ax $:

1. **Find the Eigenvalues**: Solve $\det(A - \lambda I) = 0$ to find the eigenvalues $\lambda$.

2. **Find the Eigenvectors**: For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = 0$ to find the eigenvectors $\mathbf{v}$.

3. **Construct the General Solution**: General solution will be of the form $x(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2$, where $\lambda_1, \lambda_2$ and $\mathbf{v}_1, \mathbf{v}_2$ are the eigenvalues and corresponding eigenvectors, and $c_1, c_2$ are arbitrary constants.

---

Utilize this procedure to find the solutions for the matrices provided in problems (a) and (b). If additional support or graphical descriptions are necessary, consult supplementary learning materials or a tutor.

Here is the transcription of the text from the provided image for educational purposes:

---

**(c) $ \mathbf{A} = \begin{bmatrix} -2 & -1 \\ 4 & 2 \end{bmatrix} $.**

---

In the above equation, matrix $ \mathbf{A} $ is given as a 2x2 matrix with the elements arranged in two rows and two columns. The top-left element of the matrix is -2, top-right is -1, bottom-left is 4, and bottom-right is 2.

Such matrices are commonly used in linear algebra to perform various operations such as matrix multiplication, finding determinants, and solving systems of linear equations.

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