import numpy as np
from matplotlib import pyplot as plt

def ForwardEuler(h,T):
    N = int(T/h)
    x,y = np.zeros(N),np.zeros(N)
    x[0] = 1.0
    y[0] = 0.0
    for n in range(1,N):
        x[n] = x[n-1] - h*y[n-1]
        y[n] = y[n-1] + h*x[n-1]
    return x,y

h = 0.02
T = 4.0*np.pi
x,y = ForwardEuler(h,T)

plt.plot(x,y)
plt.axes().set_aspect('equal')

def BackwardEuler(h,T):
    N = int(T/h)
    x,y = np.zeros(N),np.zeros(N)
    x[0] = 1.0
    y[0] = 0.0
    for n in range(1,N):
        x[n] = (x[n-1] - h*y[n-1])/(1.0 + h**2)
        y[n] = y[n-1] + h*x[n]
    return x,y

h = 0.02
T = 4.0*np.pi
x,y = BackwardEuler(h,T)

plt.plot(x,y)
plt.axes().set_aspect('equal')

def Trapezoid(h,T):
    N = int(T/h)
    x,y = np.zeros(N),np.zeros(N)
    x[0] = 1.0
    y[0] = 0.0
    for n in range(1,N):
        x[n] = ((1.0-0.25*h**2)*x[n-1] - h*y[n-1])/(1.0 + 0.25*h**2)
        y[n] = y[n-1] + 0.5*h*(x[n-1] + x[n])
    return x,y

h = 0.02
T = 4.0*np.pi
x,y = Trapezoid(h,T)

plt.plot(x,y)
plt.axes().set_aspect('equal')