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The problem requires solving the initial value problem (IVP) \( y' = 1 - x + 4y, \, y(0) = 1 \) using the classical Runge-Kutta method of order 4 with step sizes \( h = 0.1 \) and \( h = 0.2 \), and then using an extrapolation technique to improve the accuracy. ### Step-by-step Solution: #### 1. Classical 4th-order Runge-Kutta Method (RK4) The classical RK4 method uses the following formulas to calculate the next value \( y_{n+1} \) from the current value \( y_n \) at a step size \( h \): \[ k_1 = h \cdot f(x_n, y_n) \] \[ k_2 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right) \] \[ k_3 = h \cdot ___ _________ _______ __________ _____ ____.
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