from math import sqrt

def rk4(f, x0, y0, x1, n):
    vx = [0] * (n + 1)
    vy = [0] * (n + 1)
    h = (x1 - x0) / float(n)
    vx[0] = x = x0
    vy[0] = y = y0
    for i in range(1, n + 1):
        k1 = h * f(x, y)
        k2 = h * f(x + 0.5 * h, y + 0.5 * k1)
        k3 = h * f(x + 0.5 * h, y + 0.5 * k2)
        k4 = h * f(x + h, y + k3)
        vx[i] = x = x0 + i * h
        vy[i] = y = y + (k1 + k2 + k2 + k3 + k3 + k4) / 6
    return vx, vy

def f(x, y):
    return x * sqrt(y)

vx, vy = rk4(f, 0, 1, 10, 100)
for x, y in list(zip(vx, vy))[::10]:
    print(% (x, y, y - (4 + x * x)**2 / 16))

 0.0    1.00000  +0.0000e+00
 1.0    1.56250  -1.4572e-07
 2.0    4.00000  -9.1948e-07
 3.0   10.56250  -2.9096e-06
 4.0   24.99999  -6.2349e-06
 5.0   52.56249  -1.0820e-05
 6.0   99.99998  -1.6595e-05
 7.0  175.56248  -2.3518e-05
 8.0  288.99997  -3.1565e-05
 9.0  451.56246  -4.0723e-05
10.0  675.99995  -5.0983e-05