Solve 2 [-[-530] = x(0) = 6, y(0) = = - 22 10 1 [3] 1-18-18- 16 = +
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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![### Solving a System of Linear Differential Equations
Consider the system of linear differential equations represented as:
\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
\begin{bmatrix}
5 & 2 \\
-10 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix},
\quad x(0) = 6, \quad y(0) = -22
\]
The goal is to solve this system given the initial conditions.
On the right, the general solution to the system is represented as:
\[
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
\text{ } \\
\text{ }
\end{bmatrix}
\begin{bmatrix}
\text{ }
\end{bmatrix}
+
\begin{bmatrix}
\text{ } \\
\text{ }
\end{bmatrix}
e^{\text{ }}
\]
Where:
- The first bracket pair on the right holds constants determined by the initial conditions.
- The second bracket pair holds eigenvectors corresponding to the eigenvalues of the coefficient matrix.
- The exponential term represents the growth factor related to the eigenvalues.
These values are typically determined through the processes of finding the eigenvalues and eigenvectors of the coefficient matrix, followed by using the initial conditions to solve for any constants.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F679bd822-ca30-4f64-a182-eaf9cc9fd87a%2F80ewmij_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving a System of Linear Differential Equations
Consider the system of linear differential equations represented as:
\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
\begin{bmatrix}
5 & 2 \\
-10 & 1
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix},
\quad x(0) = 6, \quad y(0) = -22
\]
The goal is to solve this system given the initial conditions.
On the right, the general solution to the system is represented as:
\[
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
\text{ } \\
\text{ }
\end{bmatrix}
\begin{bmatrix}
\text{ }
\end{bmatrix}
+
\begin{bmatrix}
\text{ } \\
\text{ }
\end{bmatrix}
e^{\text{ }}
\]
Where:
- The first bracket pair on the right holds constants determined by the initial conditions.
- The second bracket pair holds eigenvectors corresponding to the eigenvalues of the coefficient matrix.
- The exponential term represents the growth factor related to the eigenvalues.
These values are typically determined through the processes of finding the eigenvalues and eigenvectors of the coefficient matrix, followed by using the initial conditions to solve for any constants.
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