Solve 1 0-²40 = 5 x (0) = - 4, y(0) = 10
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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![### Solve the System of Differential Equations
Given the following system of differential equations:
\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
\begin{bmatrix}
1 & 1 \\
-5 & -1
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
\]
with the initial conditions \( x(0) = -4 \) and \( y(0) = 10 \), solve for \( x \) and \( y \).
### Solution Form
The solution to the system can be expressed in the following form:
\[
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
\text{(First expression for } x \text{)}
\end{bmatrix}
+
\begin{bmatrix}
\text{(Second expression for } x \text{)}
\end{bmatrix}
\]
In this notation, the first large brackets contain the first component of the solution, and the second large brackets contain the second component of the solution.
### Explanation
In this section, the goal is to find particular solutions for \( x(t) \) and \( y(t) \) that satisfy both the differential equations and initial conditions presented. This typically involves eigenvalue analysis of the coefficient matrix, finding eigenvectors, and assembling the general solution before fitting it to the given initial conditions. The exact solutions would be filled inside the respective matrix slots.
### Graphs or Diagrams
No specific graphs or diagrams are provided in the image, but if created, they would generally depict the behavior of the solutions \( x(t) \) and \( y(t) \) over time \( t \). Detailed steps and exact solutions within the square brackets would normally be provided based on the determined eigenvalues and eigenvectors for the given system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F7ba1ad85-200c-493f-873f-c79879679465%2Fhlxdvvo_processed.png&w=3840&q=75)
Transcribed Image Text:### Solve the System of Differential Equations
Given the following system of differential equations:
\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix}
=
\begin{bmatrix}
1 & 1 \\
-5 & -1
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
\]
with the initial conditions \( x(0) = -4 \) and \( y(0) = 10 \), solve for \( x \) and \( y \).
### Solution Form
The solution to the system can be expressed in the following form:
\[
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
\text{(First expression for } x \text{)}
\end{bmatrix}
+
\begin{bmatrix}
\text{(Second expression for } x \text{)}
\end{bmatrix}
\]
In this notation, the first large brackets contain the first component of the solution, and the second large brackets contain the second component of the solution.
### Explanation
In this section, the goal is to find particular solutions for \( x(t) \) and \( y(t) \) that satisfy both the differential equations and initial conditions presented. This typically involves eigenvalue analysis of the coefficient matrix, finding eigenvectors, and assembling the general solution before fitting it to the given initial conditions. The exact solutions would be filled inside the respective matrix slots.
### Graphs or Diagrams
No specific graphs or diagrams are provided in the image, but if created, they would generally depict the behavior of the solutions \( x(t) \) and \( y(t) \) over time \( t \). Detailed steps and exact solutions within the square brackets would normally be provided based on the determined eigenvalues and eigenvectors for the given system.
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