Find the general solutions of the system. 30-4 4# -15 -1 x 10 7 x(t) =
Chapter4: Systems Of Linear Equations
Section4.6: Solve Systems Of Equations Using Determinants
Problem 279E: Explain the steps for solving a system of equations using Cramer’s rule.
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Question
![**Find the general solutions of the system.**
\[
x' = \begin{bmatrix}
3 & 0 & -4 \\
-1 & 5 & -1 \\
1 & 0 & 7
\end{bmatrix} x
\]
---
**Solution**:
The system of differential equations is represented as:
\[ x' = \mathbf{A} x \]
where,
\[ \mathbf{A} = \begin{bmatrix}
3 & 0 & -4 \\
-1 & 5 & -1 \\
1 & 0 & 7
\end{bmatrix} \]
Finding the general solution involves solving the eigenvalue problem for matrix \(\mathbf{A}\).
First, find the eigenvalues \(\lambda\) by solving the characteristic equation:
\[ \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \]
Once the eigenvalues are obtained, find the corresponding eigenvectors. Finally, combine the eigenvalues and eigenvectors to form the general solution:
\[ x(t) = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2 + c_3 e^{\lambda_3 t} v_3 \]
where \(c_1, c_2, c_3\) are constants determined by initial conditions, and \(v_1, v_2, v_3\) are the eigenvectors corresponding to the eigenvalues \(\lambda_1, \lambda_2, \lambda_3\).
This process involves linear algebra techniques and solving differential equations. The exact form of the general solution will require these steps to be carried out explicitly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F01cf19a9-a9ed-43a4-89e5-0cf8832e1c7c%2F8cp1dcc_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the general solutions of the system.**
\[
x' = \begin{bmatrix}
3 & 0 & -4 \\
-1 & 5 & -1 \\
1 & 0 & 7
\end{bmatrix} x
\]
---
**Solution**:
The system of differential equations is represented as:
\[ x' = \mathbf{A} x \]
where,
\[ \mathbf{A} = \begin{bmatrix}
3 & 0 & -4 \\
-1 & 5 & -1 \\
1 & 0 & 7
\end{bmatrix} \]
Finding the general solution involves solving the eigenvalue problem for matrix \(\mathbf{A}\).
First, find the eigenvalues \(\lambda\) by solving the characteristic equation:
\[ \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \]
Once the eigenvalues are obtained, find the corresponding eigenvectors. Finally, combine the eigenvalues and eigenvectors to form the general solution:
\[ x(t) = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2 + c_3 e^{\lambda_3 t} v_3 \]
where \(c_1, c_2, c_3\) are constants determined by initial conditions, and \(v_1, v_2, v_3\) are the eigenvectors corresponding to the eigenvalues \(\lambda_1, \lambda_2, \lambda_3\).
This process involves linear algebra techniques and solving differential equations. The exact form of the general solution will require these steps to be carried out explicitly.
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