### Differential Equations

**In this post we will discuss about "Formation of Ordinary Differential Equations"**

Let ;

**f ( x , y , c1 ) = 0 ---------------------(1)**

be an equation containing

**x , y**an on arbitrary constant

**c1 .**

Differentiating

**(1)**we get ;

**( Df/Dx ) + ( Df/Dx ) (dy/dx ) = 0 ------(2)**

equation

**(2)**will in general contains

**c1**. If

**c1**be eliminated between

**(1)**and

**(2)**, we shall get a relation involving

**x, y**and

**dy/dx**which will evidently be a differential equation of the first order .

Similarly , if we have an equation

**f ( x , y , c1 , c2 ) = 0 ----------------(3)**

containing two arbitrary constant

**c1**and

**c2**, then by differentiating this twice , we shall get two equations . Now between these two equations and given equations , in all three equations , if the two arbitrary constant

**c1**and

**c2**be eliminated , we shall evidently get a differential equation of the second order .

in general , if we have an equation ;

**f ( x , y , c1 , c2 , .......cn ) = 0 ----------(4)**

containing n arbitrary constants

**c1 , c2 , .....cn**then by differentiating this n times , we shall get

**n**equations . Now between these

**n**equations and the given equation in all

**( n+1 )**equations , if the

**n**arbitrary constants

**c1 , c2 , ...cn**be eliminated , we shall evidently get a differential equation for

**n th**order , for there being

**n**differentiations , the resulting equation must contain a derivative of the

**n th**order .

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