ABSTRACT
A function F (x , y ) is said to be algebraically homogeneous of degree n if the sum of the powers of x and y in each term of F is equal to n . Expressed
differently, F (x , y) is homogeneous of degree n if
F (tx, ty) t nF (x, y):
Thus the function
xy23x2y9x3
is homogeneous of degree 3, while the function
4 x2
2y2
is homogeneous of degree zero. However, the function
x
3y2
x3
2y2
is not homogeneous because the expression under the square root sign is not
homogeneous of degree 2, and hence homogeneous of degree 1 after the
square root operation has been performed. If a first order ordinary differential can be written as
dy
dx
M(x, y)
N(x, y) ,
where M and N are algebraically homogeneous functions of the same degree, the differential equation is said to be algebraically homogeneous or, more
simply, homogeneous.
