Consider the Diff Equ y y y3 Find the stationary points Wh

Consider the Diff Equ y\' = y - y^3 Find the stationary points Which of the stationary points are stable If y(0) = 2 use Euler\'s method with a constant h = 1/2 to approximate y(1/2) and y(1) (No Calculator is needed) Find the solution to the IVP: y\' = y - y^3 with y(0) = 2

Solution

y\' = y - y³
a) For stationary point , y\' = 0 => y - y³ = 0
   y(1-y)(1+y) = 0
   y = 0, -1, 1 are stationary points

b) let y\' = f(y) = y - y³
   f\'(y) = 1 - 3y²
   f\'(0) = 1 ; f\'(-1) = -2 ; f\'(1) = -2
for stationary points to be stable we require f\'(y) < 0
hence y= -1 and y= 1 are stable

c) Euler\'s method :
   y_n+1 = y_n + hf(t_n , y_n )
given y(0) = 0
y(1/2) = y(0) + (1/2)*f(0,0) = 2 + (1/2)(0 - 0³)= 2
y(1) = y(1/2) + (1/2)*2 = 2 +1 = 3

d) dy/dt = y(1-y)(1+y) ; y(0) = 2
dy{ 1/2(1-y) + 1/y - 1/2(1+y)} = dt
   integrating both sides and using initial value we get
   y²/(1-y²) = e^2t - 4/3