A function f(x,y) is homogeneous of degree n if f(tx,ty) = $t^n$ * f(x,y). A DE dy/dx = f$\frac{x,y}{g}$(x,y) is homogeneous when f and g have the same degree, making the ratio degree 0.

Recognition: Each term should have the same total degree in x and y. Examples: $\frac{x^2+xy}{(x^2-y^2)}$ is homogeneous (all terms degree 2). $\frac{x+y+1}{(x-y)}$ is NOT (constant term 1 breaks homogeneity).

Solving with y = vx:

1. Substitute y = vx, dy/dx = v + x$\frac{dv}{dx}$
2. Express RHS in terms of v: dy/dx = phi(v)
3. Separate: $\frac{dv}{phi(v}$-v) = $\frac{dx}{x}$
4. Integrate and substitute back v = $\frac{y}{x}$

When to use x = vy instead: If the DE is more naturally expressed as dx/dy = psi$\frac{x}{y}$, use x = vy. This gives dx/dy = v + y$\frac{dv}{dy}$ = psi(v).

Reducible to Homogeneous (Non-homogeneous linear type): dy/dx = $\frac{a1x + b1y + c1}{(a2x + b2y + c2)}$

Case 1: a1/a2 != $\frac{b1}{b2}$ — Translate: x = X+h, y = Y+k, choose h,k to make c1,c2 vanish. Case 2: a1/a2 = $\frac{b1}{b2}$ = m — Substitute v = a1x + b1y. The equation becomes separable.