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Consider the system: d = x(2-x)²(x + 3). dx dt 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.

Consider the system: d = x(2-x)²(x + 3). dx dt 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.

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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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.
Transcribed Image Text: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.

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