2. Consider the nonlinear system of differential equations dx dy = x²y – xy dt = y – e* dt (a) Determine all critical points of the system. (b) For each critical point not on the y-axis: i. Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system. ii. Find the general solution of the linearised system using eigenvalues and eigenvectors.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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2. Consider the nonlinear system of differential equations
dx
dy
= x²y – xy
dt
= y – e*
dt
(a) Determine all critical points of the system.
(b) For each critical point not on the y-axis:
i. Determine the linearisation of the system with the critical point translated to (0,0) and
discuss whether it can be used to approximate the behaviour of the non-linear system.
ii. Find the general solution of the linearised system using eigenvalues and eigenvectors.
Transcribed Image Text:2. Consider the nonlinear system of differential equations dx dy = x²y – xy dt = y – e* dt (a) Determine all critical points of the system. (b) For each critical point not on the y-axis: i. Determine the linearisation of the system with the critical point translated to (0,0) and discuss whether it can be used to approximate the behaviour of the non-linear system. ii. Find the general solution of the linearised system using eigenvalues and eigenvectors.
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