The analytic solution for the initial value problem from part (a)(ii) isy(t) =(32 t +√1 + ( 32 t)2)1/3−(32 t +√1 + ( 32 t)2)−1/3.Calculate the accuracy of the approximation obtained by the generalised Heun’s meth-ods at point t = 2.

Apply the generalised Heun’s method from part (a)(i) to solve the initial value prob-lemy′(t) = 11 + y2 , y(0) = 0.Use 2 steps of the algorithm with the step size h = 1 to approximately find y(2).Show all calculations with at least 6 accurate decimal digits. [10 marks](iii) The analytic solution for the initial value problem from part (a)(ii) isy(t) =(32 t +√1 + ( 32 t)2)1/3−(32 t +√1 + ( 32 t)2)−1/3.Calculate the accuracy of the approximation obtained by the generalised Heun’s meth-ods at point t = 2.

Consider solving an initial value problemy′(t) = f (t, y), y(0) = y0.(i) Derive the equations of the general step of the generalised Heun’s method, whichevaluates the right–hand side f (t, y) at the points t = tn and t = tn + 14 h in eachinterval [tn, tn + h].

Consider the initial value problemy′(t) = cos(y), y(0) = 0.(i) Apply 4 steps of forward Euler method with the step size h = 12 to approximately findy(2). Show at least 4 decimal places in your calculations. [10 marks](ii) Apply 2 steps of Heun’s method with the step size h = 1 to approximately find y(2).Show at least 4 decimal places in your calculations.

Let y = f(x) be the solution to the differential equation 𝑑𝑦𝑑𝑥=𝑓′(𝑥) with initial condition f(2) = 3. Selected values of f' are given in the table above. What is the approximation for f(2.4) if Euler’s method is used, starting at x = 2 with two steps of equal size?

Let f (x) = x2 −1 and I = [1, 4]. Find the right-hand side approximation R3 to S41f (x)dx.Recall that Rn is the right-endpoint approximation using n rectangles of equal width.A. 12B. 11C. 17.75D. 26E. 23