Consider the differential equation y' = f (t, y), a t b, y(a) = . a. Show that for some , where

Consider the differential equation
y' = f (t, y), a ‰¤ t ‰¤ b, y(a) = α.
a. Show that

for some ξ , where ti b. Part (a) suggests the difference method
wi+2 = 4wi+1 ˆ’ 3wi ˆ’ 2hf (ti ,wi), for i = 0, 1, . . . , N ˆ’ 2.
Use this method to solve
y' = 1 ˆ’ y, 0‰¤ t ‰¤ 1, y(0) = 0,
With h = 0.1. Use the starting values w0 = 0 and w1 = y(t1) = 1 ˆ’ eˆ’0.1.
c. Repeat part (b) with h = 0.01 and w1 = 1 ˆ’ eˆ’0.01.
d. Analyze this method for consistency, stability, and convergence.

y'(t) = 2h