. . Homogeneous Functions and Euler’s Theorem Definition . (a) Let A x y b c d F A ⊂ {( , ) | } : , we say that F is a homogeneous function on A , if there exists a constant p such that F F x y λ λ λ for all λ ∈ and sutitably restricted λ , x , y, such that ( λ λ A .
This constant p is called degree of F . (b) Let B x y z b c d u z v G B ⊂ {( , , ) | } : , we say that G is a homogeneous function on B , if there exists a constant p such that G z G x y z ( , , ) λ λ λ λ for all λ ∈ and sutitably restricted λ , x , y, z , such that ( λ λ λ z B . This constant p is called degree of G . Note: Division by any variable may occur, to avoid division by zero, we say that λ , x , y, z are sutitably restricted real numbers.
and - - Differentials and Partial Derivatives These types of functions are important in Ordinary differential equations (Chapter ). Let us consider some examples. Consider F x y ( , ) = x x y , ( , ) . Then λ λ = ( )( λ λ λ λ λ and hence F is a homogeneous function of degree .
On the other hand, G x y ( , ) = e x is not a homogeneous function because, G λ λ = e G x y λ λ λ ≠ for any λ ≠ and any p . Example . Show that F x y ( , ) = is a homogeneous function of degree . We compute λ λ = ( )( λ λ λ λ λ λ λ λ λ = F x y for all λ ∈ .
So F is a homogeneous function of degree . We state the following theorem of Leonard Euler on homogeneous functions. Theorem . (Euler's theorem - without proof) Suppose that A x y b c d F A { } ⊂ ( , ) | : .
If F is having continuous partial derivatives and homogeneous on A , with degree p , then x F x x y y F y x y pF x y x y A ∀ Suppose that B x y z b c d u z v F B