# Eulers metode for \dot x = x cos t. Vi sammenlikner numerisk og analytisk losning.

import numpy as np
import matplotlib.pyplot as plt


x0 = 1
dt = 0.1
tMax = 20

n = int(tMax/dt)
t = np.zeros(n)
dx = np.zeros(n)
x = np.zeros(n)

x[0] = x0
t[0] = 0

for i in range(n-1):
    t[i+1] = t[i] + dt
    dx[i] = np.cos(t[i])*x[i]
    x[i+1] = x[i] + dx[i]*dt

plt.plot(t,x,linewidth=2, label='Eulers metode')
plt.plot(t,x0*np.exp(np.sin(t)),linewidth=2, label='Analytisk l?sning')
plt.legend(loc='upper right')
plt.xlabel('t')
plt.ylabel('x(t)')
plt.savefig('euler.pdf')
plt.show()