2. Consider the problem y = x - y² = f(x, y), y(0) = 0. For NЄN, we divide the interval [0, 1] by the N + 1 mesh points, 0 = 20$=h£N=Nh=1, with h 1/N as the step size. Let N = 10, and consider the following numerical scheme, Yn+1 = Yn-1 +2hf (xn, Yn). (2) Here, y, is the numerical solution for y(x). (a) Estimate the local truncation error for (2) for an abstract function f(x,y) which has bounded partial derivatives up to order two with respect to the second component. 1 (b) In order to apply numerical scheme (2), one can get an approximation to y(x1) by explicit Euler scheme, and denote it as y₁. Then by applying (2), one can obtain Yj for j 2, N. Calcuate Yj for j 1, N in this manner.

Fig: 1