de2. Consider a model:dt= 1 - oscos -÷(1 + cos 0)(a) Determine the equilibrium points for this model.(b) Classify these equilibria.

We are given, dθdt=1-cosθ -13(1+cos θ) Condition of Equilibrium: dθdt=0 ⇒1-cos θ-13(1+cos θ)=0⇒1-cos…

Answered: de Consider a model: = 1- cos 0 – ÷(1 +…

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

de
2. Consider a model:
dt
= 1 - os
cos -
÷(1 + cos 0)
(a) Determine the equilibrium points for this model.
(b) Classify these equilibria.

Expert Solution

Part (a)

We are given,

$\frac{d θ}{d t} = 1 - \cos θ - \frac{1}{3} (1 + \cos θ)$

Condition of Equilibrium:

$\frac{d θ}{d t} = 0$

$\Rightarrow 1 - \cos θ - \frac{1}{3} (1 + \cos θ) = 0 \Rightarrow 1 - \cos θ = \frac{1}{3} (1 + \cos θ) \Rightarrow 3 - 3 \cos θ = 1 + \cos θ \Rightarrow 2 = 4 \cos θ \Rightarrow \cos θ = \frac{1}{2} \Rightarrow θ = \frac{π}{3}$

Hence the Equilibrium point is at $θ = \frac{π}{3}$ .

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