Explanation Given: The given autonomous differential equations are d x d t = x n and d x d t = − x n . Calculation: Consider the differential equation. d x d t = x n d t d x = 1 x n (1) Here, n is a positive integer. Find the critical points for Equation (1). 1 x n = 0 Thus, x → ∞ for n > 0 x → 0 for n ≤ 0 The differential equation is semi stable when d 2 t d x 2 < 0 . d 2 t d x 2 = d d t ( 1 x n ) = − n x n + 1 < 0 for n even Thus, the differential equation is semi stable for n even. The differential equation is unstable when d 2 t d x 2 > 0

6th Edition

ISBN: 9781284127003

Author: ZILL

Publisher: JONES+BARTLETT LEARNING,LLC (CC)

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Differential Equations

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Chapter 2, Problem 5CR

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The values of n for the given equation for which 0 is asymptotically stable, semi-stable and unstable.

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The values of n for the given equation for which 0 is asymptotically stable, semi-stable and unstable.

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Chapter 2 Solutions