Consider the system:dxdt=x (2 − x)²(x + 3).How many Stable, Unstable, Mixed (i.e. Half Stable) fixed points does this systempossess?2 unstable, 1 stable, and 1 mixed.2 stable and 2 unstable.1 unstable, 1 stable, and 1 mixed.2 mixed and 1 stable.

Answered: Consider the system: d = x(2-x)²(x +…

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

Consider the system:
dx
dt
=
x (2 − x)²(x + 3).
How many Stable, Unstable, Mixed (i.e. Half Stable) fixed points does this system
possess?
2 unstable, 1 stable, and 1 mixed.
2 stable and 2 unstable.
1 unstable, 1 stable, and 1 mixed.
2 mixed and 1 stable.

Expert Solution

Step 1

Introduction:

For the system $y' = f (x)$ . As a result, if $f' (x) > 0$ , the perturbation of the fixed point solution $x (t) = x$ decays exponentially, and we can claim that the fixed point is stable. If $f' (x) < 0$ , the fixed point is considered unstable since the perturbation rises exponentially. The fixed point is said to be marginally stable if $f' (x) = 0$ , in which case the next higher-order term in the Taylor series expansion needs to be taken into account.