calculate the numeric solution at each step to the ode fx y using rk4 https en wikipedia orgwikirungekuttamethods arguments f the ode as a function of x and y y0 the initial value for y x0 the initial value for x h the stepsize xend the end value for x the exact solution is math expx def fx y return y y0 1 y rungekuttaf y0 0 0 0 01 5 y1 148 41315904125113 calculate the numeric solution at each step to the ode f x y using rk4 https en wikipedia org wiki runge kutta_methods arguments f the ode as a function of x and y y0 the initial value for y x0 the initial value for x h the stepsize x_end the end value for x the exact solution is math exp x def f x y return y y0 1 y runge_kutta f y0 0 0 0 01 5 y 1 148 41315904125113
import numpy as np


def runge_kutta(f, y0, x0, h, x_end):
    
    n = int(np.ceil((x_end - x0) / h))
    y = np.zeros((n + 1,))
    y[0] = y0
    x = x0

    for k in range(n):
        k1 = f(x, y[k])
        k2 = f(x + 0.5 * h, y[k] + 0.5 * h * k1)
        k3 = f(x + 0.5 * h, y[k] + 0.5 * h * k2)
        k4 = f(x + h, y[k] + h * k3)
        y[k + 1] = y[k] + (1 / 6) * h * (k1 + 2 * k2 + 2 * k3 + k4)
        x += h

    return y


if __name__ == "__main__":
    import doctest

    doctest.testmod()