Solving First-order Ordinary differential equations

Solving First-order Ordinary differential equations

In this section, We discuss Ordinary Differential equations the method to solving first order Ordinary differential equations in Python Programming. The first thing we learn here is What Differential equations, a form of the differential equations. Then we moved further to solve the ODE order first and plot these equations in Python. 

In this, we used from scipy.integrate import odeint to solving first-order Ordinary differential equations. The differential equations are solved using scipy.integrate package using the function ODEINT in Python Programming. If we talk more about also need numpy and matplotlib library. Check this article to know more about these libraries

So let's start with Differential equations also discuss which form is used to write the differential equations.

Differential equations

Differential equations are called partial differential equations (PDE) or Ordinary differential equations (ODE), whether they are included or not partial derivatives. A solution (or special solution) of the differential equation of order n defines a function and has different times on one Domain D has the property that holds the functional equation obtained by substituting the function and its n derivatives into the differential equation D.

The differential equations in the form,

Equation 1.

Where f is given and is an unknown function of y. When found to be continuous over some interval, we found a general solution y(x) by integration. In section, We Solved Ordinary differential equations for the type of first order.

A first-order differential equation is an equation in which ƒ(x, y) is a function of two variables defined on a region in the XY-plane. The equation is of the first order because it contains only the first derivative dy/dx.
                                  y' = f(x,y) and    
                                   

all these are equivalent to equation 1.

Let's take a look at how to Plot first-order differential equations in Python. The list of Different equations that we solve in Python given below. We discuss these equations one by one in an easy way. 

1. Solving Ordinary differential equation of order first using Eq.

from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return x
y0 = 1xs = np.arange(0,1,0.1)
ys = odeint(f,y0,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=x, y(0)=1, xrange=[0,1]')
plt.show()

Output:

2. Solving Ordinary differential equation of order first using Eq.

from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return y/x
y1 = 1xs = np.arange(1,10,1)
ys = odeint(f,y1,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=y/x, y(1)=1')
plt.show()

3. Solving Ordinary differential equation of order first using Eq.

from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return (x*y/(np.sqrt(y**2+x)))
y0 = 1xs = np.arange(0,5,0.25)
ys = odeint(f,y0,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=(x*y/(np.sqrt(y**2+x))')
plt.show()

4. Solving Ordinary differential equation of order first using Eq.

from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return (x +y)
y0 = 0xs = np.arange(0,2,0.1)
ys = odeint(f,y0,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=x+y')
plt.show()

5. Solving Ordinary differential equation of order first using Eq.


from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return (3*y-x)
y0 = 0xs = np.arange(0,3,0.1)
ys = odeint(f,y0,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=(3*y-x)')
plt.show()




6. Solving Ordinary differential equation of order first using Eq.




from scipy.integrate import odeint
import numpy as np
from matplotlib import pyplot as plt
def f(y,x):
    return (x**2)
y0 = 1xs = np.arange(0,5,0.25)
ys = odeint(f,y0,xs)
plt.plot(xs,ys,'-')
plt.plot(xs,ys,'ro')
plt.xlabel('x values')
plt.ylabel('y values')
plt.title('Differential equation: dy/dx=(x**2)')
plt.show()