# Homogeneous Equations

The function

*f*(*x*,*y*) is said to be a homogeneous function of degree*n*if for any real number*t*(â‰ 0), we have*f*(*tx*,*ty*) =*t*^{n}*f*(*x*,*y*), e.g.,*f*(*x*,*y*) =*ax*^{2/3}+*hx*^{1/3}â‹…*y*^{1/3}+*by*^{2/3}is a homogeneous function of degree 2/3.A differential equation of the form = , where

*f*(*x*,*y*) and*Ï†*(*x*,*y*) are homogeneous functions of*x*and*y*, and of the same degree, is called*homogeneous*.This equation may also be reduced to the form = and is solved by putting

*y*=*vx,*so that the dependent variable*y*is changed to another variable*v*, where*v*is some unknown function. The differential equation is transformed to an equation with variables separable.# Equations reducible to the homogeneous form

Equations of the form (

*aB*â‰*Ab*and*A*+*b*â‰ 0) can be reduced to a homogeneous form by changing the variable*x*,*y*, to*X*,*Y*by writing*x*=*X*+*h*and*y*=*Y*+*k*; where*h*,*k*are constants to be chosen so as to make the given equation homogeneous. We have . Hence the given equation becomes .Let

*h*and*k*be so chosen as to satisfy the relation*ah*+*bk*+*c*= 0 and*Ah*+*Bk*+*C*= 0. These give*h*= , which are meaningful when

*aB*â‰

*Ab*. can now be solved by means of the substitution

*Y*=

*VX.*

In case

*aB*=*Ab*, we write*ax*+*by*=*t*. This reduces the differential equation to the separable variable type.If

*A*+*b*= 0 then a simple cross-multiplication and substituting*d*(*xy*) for*xdx*+*ydy*and integrating term by term yields the result easily.