Apply the Backward Euler method to the differential equations given in Exercise 1. Use Newton’s method to solve for w_i+1.

In Exercise 1

a. y' = −9y, 0≤ t ≤ 1, y(0) = e, with h = 0.1; actual solution y(t) = e^1−9t .

b. y' = −20(y−t²)+2t, 0≤ t ≤ 1, y(0) = 1/3 , with h = 0.1; actual solution y(t) = t²+1/3 e^−20t .

c. y' = −20y + 20 sin t + cos t, 0 ≤ t ≤ 2, y(0) = 1, with h = 0.25; actual solution y(t) = sin t + e^−20t

d. y' = 50/y−50y, 0≤ t ≤ 1, y(0) = √2, with h = 0.1; actual solution y(t) = (1+e^−100t)^1/2.