Consider the nonlinear system of differential equations dax=(2²+1) (-36-24), 1/1 = (y+¹) (2y=x²-2x) do (a) - find ALL the critical points for this system (6)- compute the Jacobi matrix and hence determine the linearisation of the system at the point (0,0). (C)-using eigenvalues and eigenvectors, find the general solution of the linearised system in part (b) at the point (0,0)

Consider the nonlinear system of differential equations dax=(2²+1) (-36-24), 1/1 = (y+¹) (2y=x²-2x) do (a) - find ALL the critical points for this system (6)- compute the Jacobi matrix and hence determine the linearisation of the system at the point (0,0). (C)-using eigenvalues and eigenvectors, find the general solution of the linearised system in part (b) at the point (0,0)

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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Question

Can someone please help with my revision.

Consider the nonlinear system of differential equations
do
da = (2+1) (-36-24), 1/1 = (y+¹) (2y=x²-2x)
(a) - find ALL the critical points for this system
(6) - compute the Jacobi matrix and hence determine the linearisation of the system
at the point (0,0).
(C)-using eigenvalues and eigenvectors, find the general solution of the linearised
system in part (b) at the point (0,0)

Expert Solution

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Step 1: Find critical points

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Step 2: Find linearization at (0,0)

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Step 3: Find eigenvalues and eigenvectors

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Step 4: Find general solution of linearized system

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Follow-up Questions

Read through expert solutions to related follow-up questions below.

Follow-up Question

Can you please answer these questions in regards to the above problem being solved.

For the linearised system in part (b):
• Find any straight line orbits and their directions;
determine the behaviour of the orbits as t→∞;
determine the behaviour of the orbits as t→∞;
• determine the slopes at which the orbits cross the coordinate axes.
Hence, sketch (by hand) a phase portrait for the linearised system in (b) around (0,0),
showing all straight line orbits and at least four other orbits, and identify the type and
stability of the critical point.
•Suppose x(0) > 0 and y(0) <0. Based on the global phase portrait, discuss what happens
to r(t) and y(t) as t→∞.

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