2. For the system of equations: dx = a11x + a12y, dt dy a21x + az2y. dt First, review lecture notes to see how we eliminate one variable to arrive at a single second-order equation. Then find the characteristic equation and solve the eigenvalues of the equation. Once x(t) is found, y(t) can be found by setting 1 dx - a11x) y(t)= (a12 # 0) a12 dt By using the above method, find solutions for the following systems: (a) dx = -4x + y, dt dy = 3x. dt
2. For the system of equations: dx = a11x + a12y, dt dy a21x + az2y. dt First, review lecture notes to see how we eliminate one variable to arrive at a single second-order equation. Then find the characteristic equation and solve the eigenvalues of the equation. Once x(t) is found, y(t) can be found by setting 1 dx - a11x) y(t)= (a12 # 0) a12 dt By using the above method, find solutions for the following systems: (a) dx = -4x + y, dt dy = 3x. dt
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![For the system of equations:
\[
\frac{dx}{dt} = a_{11}x + a_{12}y,
\]
\[
\frac{dy}{dt} = a_{21}x + a_{22}y.
\]
First, review lecture notes to see how we eliminate one variable to arrive at a single second-order equation. Then find the characteristic equation and solve the eigenvalues of the equation. Once \( x(t) \) is found, \( y(t) \) can be found by setting
\[
y(t) = \frac{1}{a_{12}}\left(\frac{dx}{dt} - a_{11}x\right) \quad (a_{12} \neq 0).
\]
By using the above method, find solutions for the following systems:
(a)
\[
\frac{dx}{dt} = -4x + y,
\]
\[
\frac{dy}{dt} = 3x.
\]
(b)
\[
\frac{dx}{dt} = 2x - 3y,
\]
\[
\frac{dy}{dt} = x - 2y.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11d9adce-31d3-450e-a738-16d52e01b365%2Facf8e012-9cb1-4118-9571-434486975b16%2Fxia99_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For the system of equations:
\[
\frac{dx}{dt} = a_{11}x + a_{12}y,
\]
\[
\frac{dy}{dt} = a_{21}x + a_{22}y.
\]
First, review lecture notes to see how we eliminate one variable to arrive at a single second-order equation. Then find the characteristic equation and solve the eigenvalues of the equation. Once \( x(t) \) is found, \( y(t) \) can be found by setting
\[
y(t) = \frac{1}{a_{12}}\left(\frac{dx}{dt} - a_{11}x\right) \quad (a_{12} \neq 0).
\]
By using the above method, find solutions for the following systems:
(a)
\[
\frac{dx}{dt} = -4x + y,
\]
\[
\frac{dy}{dt} = 3x.
\]
(b)
\[
\frac{dx}{dt} = 2x - 3y,
\]
\[
\frac{dy}{dt} = x - 2y.
\]
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