from pylab import plot,show,ylabel,xlabel

def expeuler(N,T,t0=0.0,u0 =0.0):
    dt = 1.*T/N
    t = [t0,]
    u = [u0,]
    
    for i in range(N):
        u +=[u[-1] + dt*(u[-1] + t[-1]*(2. -t[-1]))]
        t += [t[-1]+dt]
    return u,t


def impeuler(N,T,u0=0.0,t0=0.0):
    dt= 1.*T/N
    t = [t0,]
    u = [u0,]
    for i in range(N):
        u +=[(u[-1] +dt*t[-1]*(2. - t[-1]))/(1. - dt)]
        t += [t[-1]+dt] 
    return u,t
s = expeuler(20,1.)

plot(s[1],s[0])    
s = impeuler(20,1.)
plot(s[1],s[0])

show()