. . Homogeneous Form or Homogeneous Differential Equation Definition . : (Homogeneous Function of degree n ) A function f x y ( , ) is said to be a homogeneous function of degree n in the variables x and y if, f tx ty t f x y ( , for some n ∈ for all suitably restricted x y and t .
This is known as Euler’s homogeneity . For instance, (i) f x y ( , ) = is a homogeneous function in x and y , of degree two. (ii) But f x y x e y + ( is not a homogeneous function. If f x y ( , ) is a homogeneous function of degree zero, then there exists a function g such that f x y ( , ) is always expressed in the form g or g .
Definition . : (Homogeneous Differential Equation) An ordinary differential equation is said to be in homogeneous form , if the differential equation is written as dy g . Caution The word “homogeneous” used in Definition . is different from in Definition .
. Remark (i) The differential equation M x y dx N x y dy = [in differential form] is said to be homogeneous if M and N are homogeneous functions of the same degree . (ii) The above equation is also written as dy f x y ( , ) [in derivative form] where f x y M x y N x y ( , ) / = − is clearly homogeneous of degree . For instance ( ) consider the differential equation x xy dy .
The given equation is rewritten as − y x / . Thus, the given equation is expressed as y x g − = / . Hence, xy dy is a homogeneous differential equation. Ordinary Differential Equations ( ) However, the differential equation dy is not homogeneous.
(verify!) To find the solution of