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Differential EquationsEasyNumerical Methods: Euler to Runge–Kutta

Numerical Methods: Euler to Runge–Kutta · Easy problem

Apply one step of the improved Euler (Heun) method with h=12h = \dfrac{1}{2} to y′=2ty' = 2t, y(1)=4y(1) = 4. Heun's step is y1=y0+h2(k1+k2)y_1 = y_0 + \dfrac{h}{2}\left(k_1 + k_2\right) with k1=f(t0,y0)k_1 = f(t_0, y_0) and k2=f(t0+h, y0+hk1)k_2 = f(t_0 + h,\ y_0 + h k_1). What is the estimate of y(32)y\left(\dfrac{3}{2}\right)? Express your answer as a common fraction.