import numpy as np
import matplotlib.pyplot as plt


#  x' = y
#  y' = - x - a * y

T_s = 0
T_f = 20
h = 0.01
a = 0.5
x_init = 1
y_init = 0

t = np.arange(T_s, T_f, h)
x = np.zeros(t.shape)
y = np.zeros(t.shape)
x[0] = y_init
y[0] = x_init

for i in range(t.size-1):
    x[i+1] = x[i] + h * y[i]
    y[i+1] = y[i] + h * (-x[i] - a * y[i])


plt.plot(t, x, t, y, markersize=3)
plt.show()