Deduction Form of Euler ’ s Theorem If f ( x , y ) is a homogeneous function in x , y of degree n , then ( i ) x ( ∂ 2 f / ∂x 2 ) + y ( ∂ 2 f / ∂x ∂y ) = ( n – 1 ) & partf / ∂x ( ii ) x ( ∂ 2 f / ∂y ∂x ) + y ( ∂ 2 f / ∂y 2 ) = ( n – 1 ) & partf / ∂y ( iii ) x 2 ( ∂ 2 f / ∂x 2 ) + 2xy ( ∂ 2 f / ∂x ∂y ) + y 2 ( ∂ 2 f / ∂y 2 ) = n ( n – 1 ) f ( x , y ) Important Points to be Remembered
If α is m times repeated root of the equation f ( x ) = 0 , then f ( x ) can be written as f ( x ) =( x – α ) m g ( x ), where g ( α ) ≠ 0 .
From the above equation , we can see that f ( α ) = 0 , f ’ ( α ) = 0 , f ” ( α ) = 0 , … , f ( m – l ) ,( α ) = 0 . Hence , we have the following proposition f ( α ) = 0 , f ’ ( α ) = 0 , f ” ( α ) = 0 , … , f ( m – l ) ,( α ) = 0 . Therefore , α is m times repeated root of the equation f ( x ) = 0 .