Find the equilibrium solutions of the ordinary differential equationy' = x^2 cos^3 yWhich of these solutions are stable?

Answer:[tex]Y_{n}[/tex]=π/2 + nπ =(π/2)*(1+2n)    where n∈Z (integers)if n=even, Y is stableif n=odd, Y is unstableStep-by-step explanation:y' = x^2 cos^3 yEquilibrium solutions occur when y'(y,x) =f(y,x)=0, we need to find y=Y=constant and satisfy this.[tex]x^{2} cos^{3}(y)=0[/tex] ⇒ [tex]cos^{3}(y)=0[/tex] ⇒ [tex]cos(y)=0[/tex][tex]Y_{n}[/tex]=π/2 + nπ =(π/2)*(1+2n)    where n∈Z (integers)Stability analisys, f(y,x)= [tex]x^{2} cos^{3}(y)=0[/tex] (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)= [tex]x^{2} cos^{3}(y)=0[/tex] , we can verify that:if n=even, Y is stableif n=odd, Y is unstable