### 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    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 .