# 13.3.4
from numpy import *;
import matplotlib.pyplot as plt;


def euler(f, a, b, x0, n):
	x = zeros(n+1);
	x[0] = x0;
	h = (b-a)/n;
	t = linspace(a,b, n+1);
	for k in range(n):
		x[k+1] = x[k] + h*f(t[k], x[k]);
	return x;
	
def f(t, x):
	return x;

a = 0;
b = 1;
n = 10;
x0 = 1;

t = linspace(a,b,n+1);
x_euler = euler(f,a,b,x0,n);

plt.plot(t, x_euler, label="Euler");
plt.plot(t, exp(t), label = "Exact");
plt.legend();
plt.show()