import numpy as np
import matplotlib.pyplot as plt
from math import log

# Set up parameters and initial value
N0 = 100
lamb = 0.5 * log(2)

# Parameters for Euler's method
h = 1
t_final = 10

# Compute time grid and number of steps needed
t = np.arange(0, t_final, h)
steps = len(t) - 1

# Initialize solution vector
N = np.zeros((steps + 1,1))
N[0] = N0

# Use Euler's method.
# Each iteration computes a new value for N
for k in range(steps):
    N[k + 1] = N[k] - lamb * h * N[k]

# This is the exact solution
N_exact = N0 * np.exp(-lamb*t)

# Do some plotting

plt.plot(t, N, label = 'Simulated')
plt.plot(t, N_exact, label = 'Exact')
plt.legend()
plt.xlabel('t')
plt.ylabel('N(t)')
plt.show()