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Find the equilibrium solutions of the ordinary differential equation

y' = x^2 cos^3 y

Which of these solutions are stable?

1 Answer

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Answer:


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

Find the equilibrium solutions of the ordinary differential equation y' = x^2 cos-example-1
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