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Differential EquationsEasyNumerical Methods: Euler to Runge–Kutta

Numerical Methods: Euler to Runge–Kutta · Easy problem

To use Euler's method on y′′=t−yy'' = t - y, y(0)=1y(0) = 1, y′(0)=2y'(0) = 2, write it as the system u′=vu' = v, v′=t−uv' = t - u with u=yu = y, v=y′v = y'. Take one Euler step with h=12h = \dfrac{1}{2}. What is the resulting estimate of y′(12)y'\left(\dfrac{1}{2}\right)? Express your answer as a common fraction.