## 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.