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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
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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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) \[ \frac{dx}{dt} = 2x - 3y, \] \[ \frac{dy}{dt} = x - 2y. \]
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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