Solve the given system of differential equations by systematic elimination. dx 8x + 13y %3D dt dy = x – 4y dt (x(t), y(t)) = %3D
Solve the given system of differential equations by systematic elimination. dx 8x + 13y %3D dt dy = x – 4y dt (x(t), y(t)) = %3D
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 Systems of Differential Equations by Systematic Elimination
To solve the given system of differential equations by systematic elimination, follow these steps:
Given:
\[
\frac{dx}{dt} = 8x + 13y
\]
\[
\frac{dy}{dt} = x - 4y
\]
Our goal is to find the functions \( x(t) \) and \( y(t) \).
These are two linear differential equations that can be written in matrix form:
\[
\begin{pmatrix}
\frac{dx}{dt} \\
\frac{dy}{dt}
\end{pmatrix}
=
\begin{pmatrix}
8 & 13 \\
1 & -4
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
To solve this system, we need to find the eigenvalues and eigenvectors of the coefficient matrix:
\[
A =
\begin{pmatrix}
8 & 13 \\
1 & -4
\end{pmatrix}
\]
The solution \( (x(t), y(t)) \) will be expressed in terms of exponential functions involving these eigenvalues and eigenvectors.
### Final Solution Form:
\[
(x(t), y(t)) = \left( \text{Solution involving } e^{\lambda_1 t} \text{ and } e^{\lambda_2 t} \text{ with appropriate coefficients based on initial conditions and eigenvalues/eigenvectors} \right)
\]
Details of computing the eigenvalues, eigenvectors, and forming the general solution will follow the standard procedures used for linear systems of differential equations. Here, we leave the verbal form of the solution unspecified as it depends on specific computations that will be done step-by-step in the learning module.
*Note:* The actual solution process involves solving a characteristic polynomial to find eigenvalues (\(\lambda_1\) and \(\lambda_2\)), and then using these eigenvalues to find eigenvectors, which leads to the general solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ef76dc7-4187-4f97-9d11-35100314604d%2Ff39f056e-4d95-4656-8f8e-faade3b953e9%2F1s9xw9g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Systems of Differential Equations by Systematic Elimination
To solve the given system of differential equations by systematic elimination, follow these steps:
Given:
\[
\frac{dx}{dt} = 8x + 13y
\]
\[
\frac{dy}{dt} = x - 4y
\]
Our goal is to find the functions \( x(t) \) and \( y(t) \).
These are two linear differential equations that can be written in matrix form:
\[
\begin{pmatrix}
\frac{dx}{dt} \\
\frac{dy}{dt}
\end{pmatrix}
=
\begin{pmatrix}
8 & 13 \\
1 & -4
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]
To solve this system, we need to find the eigenvalues and eigenvectors of the coefficient matrix:
\[
A =
\begin{pmatrix}
8 & 13 \\
1 & -4
\end{pmatrix}
\]
The solution \( (x(t), y(t)) \) will be expressed in terms of exponential functions involving these eigenvalues and eigenvectors.
### Final Solution Form:
\[
(x(t), y(t)) = \left( \text{Solution involving } e^{\lambda_1 t} \text{ and } e^{\lambda_2 t} \text{ with appropriate coefficients based on initial conditions and eigenvalues/eigenvectors} \right)
\]
Details of computing the eigenvalues, eigenvectors, and forming the general solution will follow the standard procedures used for linear systems of differential equations. Here, we leave the verbal form of the solution unspecified as it depends on specific computations that will be done step-by-step in the learning module.
*Note:* The actual solution process involves solving a characteristic polynomial to find eigenvalues (\(\lambda_1\) and \(\lambda_2\)), and then using these eigenvalues to find eigenvectors, which leads to the general solution.
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