Question

1. (10 points) Consider the autonomous equation dy = y2 + 3y + 2 dc (a) (6 points) Determine the equilibrium solutions of the equation, and classify each as asymptotically stable or unstable. (b) (4 points) Sketch at least three solutions to the equation, choosing initial points not corresponding to the equilibrium solutions. Include the equilibrium solutions in your sketch.

Answer 1

1. y² + 37+2 f(y) 2 (say) dx a) given by, The equilibrium solutions are f(x)= 0 = y3y +2 = 0 → (+2)(4+1) = 0 => y=-1, y = -2 equilibrium solutions :. are Y = -1, Y = -2 f(y) = 2y + 3 :: f'(-1) = -2+3 = 1 >0 is unstable ..Y--1 f'(-2) - 4+ -1 <0 si Y = -2 is asymptotically stable. b) The sketch of solutions gwen below:

the red solution curve is the equilibrium solution y=-2 and the green is y=-1

3.0 2.0 1.0 / 0.0 //////// /// -1.0 -2.0 -3.0 -4 -3 -2 -1 0 1 2 3 4

The orange solution curve is for initial point y(0)=-1.5

The purple solution curve is for initial point y(0)= -0.5

The blue solution curve is for initial point y(0)= 0