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Differential EquationsHardNumerical Methods: Euler to Runge–Kutta

Numerical Methods: Euler to Runge–Kutta · Hard problem

Use two steps of the improved Euler (Heun) method with h=12h = \dfrac{1}{2} on y′=t−yy' = t - y, y(0)=1y(0) = 1, to estimate y(1)y(1). (Heun: k1=f(tn,yn)k_1 = f(t_n, y_n), k2=f(tn+h,yn+hk1)k_2 = f(t_n + h, y_n + h k_1), yn+1=yn+h2(k1+k2)y_{n+1} = y_n + \dfrac{h}{2}(k_1 + k_2).) Express your answer as a common fraction.