We have to understand about Homogeneous Equation in Calculus .
If M and N of the equation Mdx + Ndy = 0 are both of the same degree in x and y , and are homogeneous , the equation is said to be homogeneous . Such an equation can be put in the form
dy/dx = f ( y/x )
Every homogeneous equation of the above type can be easily solved by putting y = vx
where v is a function of x , and consequently
dy/dx = v + x ( dv/dx )
whereby it reduced to the form v + xdv/dx = f ( v )
i.e. dx/x = dv/ [ f(v) - v ]
in which the variables are separated
A Special Form :-
The equation of the form
dy/dx = [ ( a1x +b1y +c1 ) / ( a2x + b2y +c2 ) ]
where , [ a1/a2 is not equal to b1/b2 ] --------------(1)
can be easily solved by putting
x = x' + h
and y = y' + k
where h and k are constant
So that , dx = dx' and dy = dy'
and choosing h and k in such a way that
a1h + b1k + c1 = 0 and a2h + b2k + c2 =0 -------(2)
For now the equation reduces to the form
dy'/dx' = ( a1x' +b1y' ) / ( a2x' + b2y' )
which is homogeneous in x' and y' and hence solved .