homogeneous (same-degree) equation

Some equations look hopeless until you notice they only care about the RATIO of y to x, not the two values separately. Picture zooming a photograph in and out: if the slope pattern of the equation looks identical at every zoom level — the same on a tiny scale near the origin as far away — then the equation has a special self-similar structure. That is a homogeneous equation, and it is the second great substitution trick of first-order theory.

Precisely, dy/dx = f(x, y) is called homogeneous when f depends on x and y only through the combination y/x; equivalently, in the form M dx + N dy = 0, both M and N are homogeneous functions of the same degree, meaning M(tx, ty) = t^k M(x, y) and likewise for N. Because the right-hand side is really a function of the single ratio v = y/x, you substitute y = v x (so dy/dx = v + x dv/dx). This always converts the equation into a separable equation in v and x, which you then solve by separation of variables and finally undo with v = y/x.

Beware the word 'homogeneous': in this first-order context it means same-degree / depends only on y/x, which is completely different from the later sense of 'homogeneous linear equation' (right-hand side equal to zero). Same word, two unrelated meanings — a classic source of confusion that is worth flagging out loud.