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.