Solve Differential Equations with ODEINT Function of SciPy module in Python

Last Updated : 09 Oct, 2022

In this post, we are going to learn how to solve differential equations with odeint function of scipy module in Python.

ODE stands for Ordinary Differential Equation and refers to those kinds of differential equations that involve derivatives but no partial derivatives. In other words, we only consider one independent variable in these equations.

So, in this article we have used scipy, NumPy, and Matplotlib modules of python which you can install with the following command:

pip install scipy numpy matplotlib

The syntax of odeint functions is as follows:

odeint(func, y0, t, …..)

Parameters :
model– the differential equation
y0– Initial value of Y
t– the time space for which we want the curve(basically the range of x)

A lot more other parameters are there which you can check out in the documentation but these three parameters are the main ones.

Solving Differential Equations

Now let us solve some ODE with the help of the odeint function.

Example 1:

Ordinary Differential Equation

Python3

import numpy as np 
from scipy.integrate import odeint 
import matplotlib.pyplot as plt 
  
def returns_dydt(y,t): 
    dydt = -y * t + 13
    return dydt 
  
# initial condition 
y0 = 1
  
# values of time 
t = np.linspace(0,5) 
  
# solving ODE 
y = odeint(returns_dydt, y0, t) 
  
# plot results 
plt.plot(t,y) 
plt.xlabel("Time") 
plt.ylabel("Y") 
plt.show()

Output:

Graph for the solution of ODE

Example 2:

Ordinary Differential Equation

Python3

import numpy as np 
from scipy.integrate import odeint 
import matplotlib.pyplot as plt 
  
def returns_dydt(y,t): 
    dydt = 13 * np.exp(t) + y 
    return dydt 
  
# initial condition 
y0 = 1
  
# values of time 
t = np.linspace(0,5) 
  
# solving ODE 
y = odeint(returns_dydt, y0, t) 
  
# plot results 
plt.plot(t,y) 
plt.xlabel("Time") 
plt.ylabel("Y") 
plt.show()

Output:

Graph for the solution of ODE

Example 3:

Ordinary Differential Equation

Let us try giving multiple initial conditions by making y0 an array.

Python3

import numpy as np 
from scipy.integrate import odeint 
import matplotlib.pyplot as plt 
  
def returns_dydt(y,t): 
    dydt = (1-y)/(1.95-y) - y/(0.05+y) 
    return dydt 
  
# initial conditions 
y0 = [0, 1, 2] 
  
# values of time 
t = np.linspace(1,10) 
  
# solving ODE 
y = odeint(returns_dydt, y0, t) 
  
# plot results 
plt.plot(t,y) 
plt.xlabel("Time") 
plt.ylabel("Y") 
plt.show()