Question

Find the equilibrium solutions of the ordinary differential equation

y' = x^2 cos^3 y

Which of these solutions are stable?

Answer 1

`Y_(n)`=π/2 + nπ =(π/2)*(1+2n) where n∈Z (integers)

if n=even, Y is stable

if n=odd, Y is unstable

Explanation:

y' = x^2 cos^3 y

Equilibrium solutions occur when y'(y,x) =f(y,x)=0, we need to find y=Y=constant and satisfy this.

`x^(2) cos^(3)(y)=0` ⇒
`cos^(3)(y)=0` ⇒
`cos(y)=0`

`Y_(n)`=π/2 + nπ =(π/2)*(1+2n) where n∈Z (integers)

Stability analisys, f(y,x)=
`x^(2) cos^(3)(y)=0`

(i.) If f(y,x)) < 0 on the left of Y, and f(y) > 0 on the right of Y, then the equilibrium solution is unstable.

(ii.) If f(y,x) > 0 on the left of Y, and f(y) < 0 on the right of Y, then the equilibrium solution y = c is stable.

(iii.) If f(x) > 0 on both sides of X, or f(x) < 0 on both sides of c, then the equilibrium solution y = X is semi-stable.

So, In the graph annexed we see f(y,x)=
`x^(2) cos^(3)(y)=0` , we can verify that:

if n=even, Y is stable

if n=odd, Y is unstable