A computer will be required for the remaining problems in this section. In Problems 9 through 12, an initial value problem and its exact solution are given. In each of these four problems, use the Runge–Kutta method with step sizes h = 0.1 and h = 0.05 to approximate to five decimal places the values x(1) and y(1). Compare the approximations with the actual values.

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x' = x + 2y, y' = x + et, x(0) = 0, y(0) = 0; x (t) = (2e²t - 2e-¹ + 6te-¹), y(t)=(e2t-e-t + 6te-¹)