Question

The number 0 is a critical point of the autonomous differential equation dx/dt = 7xn, where n is a positive integer. For what values of n is 0 asymptotically stable? Semi-stable? Unstable?

Answer 1

a) 0 is stable when n = odd

b) 0 is semi-stable when n = even

c) 0 is unstable when n is odd

Explanation:

Th differential equation for this question

dx/dt = x^n

n = positive integer

a) value of n where 0 is stable

0 is stable when x^n is replaced with -x^n

because considering n to be an odd number

-x^n > 0 when x < 0 while -x^n < 0 when x > 0

∴ In this scenerio we can conclude that 0 is stable when n = odd number

b) Value of n where 0 is Semi-stable

assuming n is an even number

x^n > 0 for all the values of x

c) Value of n where 0 is unstable

lets assume n is odd

when n < 0, xⁿ < 0

when n > 0, xⁿ > 0

i.e. 0 is asymptotically unstable when n is an odd number