Python Program to Solve ODE's

With the initial conditions are; y(t=0)=4  and  k=0.1

The Python code first imports the needed Numpy, Scipy, and Matplotlib packages. The model, initial conditions, and time points are defined as inputs to scipy.integrate to numerically calculate y(t). 

import numpy as np

from scipy.integrate import odeint

import matplotlib.pyplot as plt

# function that returns dy/dt

def model(y,t,k):

dydt = -k * y

return dydt

# initial condition of y at t=0

y0 = 4

# time points total 30 points taken

t = np.linspace(0,30)

# solve the ordinary differential equation for various values of k

k = 0.1

y1 = odeint(model,y0,t,args=(k,))

k = 0.2

y2 = odeint(model,y0,t,args=(k,))

k = 0.3

y3 = odeint(model,y0,t,args=(k,))

k = 0.4

y4 = odeint(model,y0,t,args=(k,))

k = 0.5

y5 = odeint(model,y0,t,args=(k,))

# plot the results in the same plot

plt.plot(t,y1,'r-',linewidth=2,label='k=0.1')

plt.plot(t,y2,'b--',linewidth=2,label='k=0.2')

plt.plot(t,y3,'c--',linewidth=2,label='k=0.3')

plt.plot(t,y4,'m--',linewidth=2,label='k=0.4')

plt.plot(t,y5,'g:',linewidth=2,label='k=0.5')

plt.xlabel('time')

plt.ylabel('y(t)')

plt.legend()

plt.show()