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Differential EquationsHardNumerical Methods: Euler to Runge–Kutta

Numerical Methods: Euler to Runge–Kutta · Hard problem

The problem y′=1+y2y' = 1 + y^2, y(0)=0y(0) = 0 has exact solution y=tan⁡ty = \tan t. Apply one step of Heun's method (k1=f(t0,y0)k_1 = f(t_0, y_0), k2=f(t0+h, y0+hk1)k_2 = f(t_0 + h,\ y_0 + hk_1), y1=y0+h2(k1+k2)y_1 = y_0 + \dfrac{h}{2}(k_1 + k_2)) with h=12h = \dfrac{1}{2}. Express the estimate of y(12)y\left(\dfrac{1}{2}\right) as a common fraction.