import numpy as np
N = np.array([[0.436,-0.281, 0.121], \
              [0.614, 0.064, 0.0461], \
              [0.603, 0.230, 0.167]])

time = np.array([0.0, \
                 0.5-np.sqrt(5)/10.0, \
                 0.5+np.sqrt(5)/10.0, \
                 1.0])

from scipy.optimize import fsolve

y0 = 5
zGuess = np.zeros(6) # y1-y3 and dydt1-dydt3

def myFunction(z):
    y  = z[0:3]
    dy = z[3:6]

    F = np.empty(6)
    F[0:3] = np.dot(N,dy) - (y-y0)
    F[3:7] = dy + y
    return F

z = fsolve(myFunction,zGuess)
print(z)