3. Nonlinear systems of coupled 1st-order ODEs. Consider the system of coupled ODES: y - x², (a +11)(b +1)x+ (a + 11)y + (b + 1)x² + xy . x' y' = = (i) Find the critical points (xo, yo) E R2 of this system. Hint: One critical point is (0,0), and there are two more critical points. (ii) For each critical point that is not of the form (0,0), find the approximate linear ODE system that is valid in a small neighbourhood of it. (iii) Find the eigenvalues and the corresponding eigenvectors of each of the linear systems found in part (ii). Hint: You should find that all eigenvalues and eigenvectors are real.
3. Nonlinear systems of coupled 1st-order ODEs. Consider the system of coupled ODES: y - x², (a +11)(b +1)x+ (a + 11)y + (b + 1)x² + xy . x' y' = = (i) Find the critical points (xo, yo) E R2 of this system. Hint: One critical point is (0,0), and there are two more critical points. (ii) For each critical point that is not of the form (0,0), find the approximate linear ODE system that is valid in a small neighbourhood of it. (iii) Find the eigenvalues and the corresponding eigenvectors of each of the linear systems found in part (ii). Hint: You should find that all eigenvalues and eigenvectors are real.
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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PLEASE ANSWER ALL PARTS OF QUESTION 3 CLEARLY
Transcribed Image Text:3.
Nonlinear systems of coupled 1st-order ODEs. Consider the system of coupled ODES:
y - x²,
(a +11)(b +1)x+ (a + 11)y + (b + 1)x² + xy .
x'
y'
=
=
(i) Find the critical points (xo, yo) E R2 of this system.
Hint: One critical point is (0,0), and there are two more critical points.
(ii) For each critical point that is not of the form (0,0), find the approximate linear ODE
system that is valid in a small neighbourhood of it.
(iii) Find the eigenvalues and the corresponding eigenvectors of each of the linear systems
found in part (ii).
Hint: You should find that all eigenvalues and eigenvectors are real.
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