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Differential EquationsHardNumerical Methods: Euler to Runge–Kutta

Numerical Methods: Euler to Runge–Kutta · Hard problem

The problem y′=y2y' = y^2, y(0)=1y(0) = 1 has exact solution y=11−ty = \dfrac{1}{1 - t}. Use two Euler steps with h=14h = \dfrac{1}{4} to estimate y(12)y\left(\dfrac{1}{2}\right), and find the error (exact value minus Euler value). Express your answer as a common fraction.