For first order differential equations, an equation is said to be "separable" (what you are calling "variable separable) if it can be written in the form or, equivalently so that each side can be integrated separately. A first order differential equation, of the form [tex]\frac{dy}{dx}= f(x,y)[/quote] is said to be "homogeneous" if replacing x and y in f(x,y) by cx and cy respectively does not change the equation: that is, if the "c" cancels out.

"Equation of the form non variable separable, homogeneous are identical!." is not true. For example, is a "non variabel separable equation" that is not homogenous. The important thing about homogeneous equations is that they can be changed to separable equations.

is homogeneous (since (cx+cy)/(cy- cx)= c(x+y)/c(y-x)= (x+y)/(y-x)) but not separable. What is true is that if we introduce the new dependent variable v= y/x, so that y= xv, dy/dx= x dv/dx+ v, then dividing both numerator and denominator by x gives (1+ y/x)/(y/x- 1)= (1+ v)/(v- 1) so the equation becomes x dv/dx+ v= (1+v)/(v-1) which IS separable: every homogeneous equation can be changed to a separable equation by that substitution.

(Caution: Unfortunately, the word "homogeneous" is also used in a very different sense for differential equations of higher order.)