Given the initial-value problem

y' = 1/t² – y/t − y², 1≤ t ≤ 2, y(1) = −1,

With exact solution y(t) = −1/t:

a. Use Taylor’s method of order two with h = 0.05 to approximate the solution, and compare it with the actual values of y.

b. Use the answers generated in part (a) and linear interpolation to approximate the following values of y, and compare them to the actual values.

i. y(1.052)

ii. y(1.555)

iii. y(1.978)

c. Use Taylor’s method of order four with h = 0.05 to approximate the solution, and compare it with the actual values of y.

d. Use the answers generated in part (c) and piecewise cubic Hermite interpolation to approximate the following values of y, and compare them to the actual values.

i. y(1.052)

ii. y(1.555)

iii. y(1.978)