The function y(t) satisfies the ordinary differential equation dy = f(t, y) for t> 0. dt (4) The variable t takes discrete values, t with a constant step-size h = tn+1 − tn, for all n Є N. The approximation of y(tn) is denoted yn. (a) Derive an implicit scheme by integrating (4) over the interval [tn,tn+1], using the trapezium quadrature rule. [5] (b) Consider the case f(y) -Ay where > 0 is a constant. == (i) Find the exact solution of equation (4), subject to the initial condition y(0) = yo, and identify the behaviour of the solution as t→ +∞. [2] (ii) Write down the difference equation corresponding to the implicit scheme derived in part (a) and show that Yn+1 = 1-Xh/2 1+Xh/2 Yn. .[3] (iii) Solve the difference equation (i.e. write down y in terms of yo) and show that this implicit scheme is unconditionally stable. [5]