In your Week 9 large class, you considered the initial value problem of an object falling from rest, described by the differential equation M) = g wt). (1) where :t(t) 2 U is the displacement in metres of the object at t seconds, 33(0) = 0, g = 9.81 mjs2 is acceleration due to gravity, and k > [J is a constant. In Question 2 of this assignment, you solve the initial value problem analytically. In Question 3, you solve it numerically. 2. (a) Let v(t) = :r'(t). Use this substitution to transform the dierential equation (1} into another differ ential equation (2). Consider the ansatz Mt} = AeM + B. Find B such that 'u[t) satises the dierential equation (2). Using the parameter B determined in part (b), nd A such that the ansatz Mt) = Arakt +B satises the initial value problem. Find 33(t], given 13(0) = 0. Suppose we divide our time interval into subintervals of equal length At = ti+1 t5. What is the formula for the linear approximation of u{ti+1) about t = t1? Your answer may include the term v(t,-) and the parameters It and At, but may not include the term lm). What is the formula for the linear approximation of :r{t,;+1) about t = ti? Your answer may include the terms 3:3,) and Mtg), and the parameters 1:: and At. Write an Excel program to use the approximations in part (a) and part (b) to estimate M2) and 3(2), using At = 0.1 s and k = 2 s'l. You do not need to include your program in your solutions only your estimates for 13(2) and 113(2) to two decimal places