A hand-held calculator will suffice for Problems 1 through 8. In each problem an initial value problem and its exact solution are given. Approximate the values of x (0.2) and y(0.2) in three ways:

(a) By the Euler method with two steps of size h = 0.1;

(b) by the improved Euler method with a single step of size h = 0.2; and

(c) By the Runge-Kutta method with a single step of size h = 0.2.

Compare the approximate values with the actual values x (0.2) and y(0.2).

x' = 2x - 5y, x (0) = 2, y' = 4x - 2y, y (0) = 3; x (t) = 2 cos 4t - sin 4t, y(t) = 3 cos 4t + sin 4t 4