**Solving Systems of Linear Differential Equations Using Eigenvalues**The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system.\[x'_1 = 6x_1 + 5x_2 + 5x_3, \quad x'_2 = -7x_1 - 6x_2 - 5x_3, \quad x'_3 = 7x_1 + 7x_2 + 6x_3\]---**Problem Statement**What is the general solution in matrix form?\[\mathbf{x}(t) = \quad \boxed{\phantom{answer}}\]To find the general solution:1. **Form the Coefficient Matrix**: Construct the matrix A of the system of differential equations.2. **Find Eigenvalues**: Determine the eigenvalues of matrix A.3. **Find Eigenvectors**: Calculate the eigenvectors corresponding to each eigenvalue.4. **Construct General Solution**: Form the general solution using the eigenvalues and eigenvectors.By completing these steps, you will derive the general matrix form solution for the system provided.

Given system is x1'=6x1+5x2+5x3, x2'=-7x1-6x2-5x3, x3'=7x1+7x2+6x3 To Find: Eigenvalues and then…

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 6x₁ +5x₂ + 5x3, x2 = -7x₁ - 6x₂ - 5x3, x'3 = 7x₁ + 7x₂ + 6x3 What is the general solution in matrix form? x(t) =

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 6x₁ +5x₂ + 5x3, x2 = -7x₁ - 6x₂ - 5x3, x'3 = 7x₁ + 7x₂ + 6x3 What is the general solution in matrix form? 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

**Solving Systems of Linear Differential Equations Using Eigenvalues**

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system.

\[
x'_1 = 6x_1 + 5x_2 + 5x_3, \quad x'_2 = -7x_1 - 6x_2 - 5x_3, \quad x'_3 = 7x_1 + 7x_2 + 6x_3
\]

---

**Problem Statement**

What is the general solution in matrix form?

\[
\mathbf{x}(t) = \quad \boxed{\phantom{answer}}
\]

To find the general solution:

1. **Form the Coefficient Matrix**: Construct the matrix A of the system of differential equations.
2. **Find Eigenvalues**: Determine the eigenvalues of matrix A.
3. **Find Eigenvectors**: Calculate the eigenvectors corresponding to each eigenvalue.
4. **Construct General Solution**: Form the general solution using the eigenvalues and eigenvectors.

By completing these steps, you will derive the general matrix form solution for the system provided.

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