A function f(x,y) is homogeneous of degree n if f(tx,ty) = t^n * f(x,y). A DE dy/dx = f(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: (x^2+xy)/(x^2-y^2) is homogeneous (all terms degree 2). (x+y+1)/(x-y) is NOT (constant term 1 breaks homogeneity).
Solving with y = vx:
- Substitute y = vx, dy/dx = v + x(dv/dx)
- Express RHS in terms of v: dy/dx = phi(v)
- Separate: dv/(phi(v)-v) = dx/x
- Integrate and substitute back v = y/x
When to use x = vy instead: If the DE is more naturally expressed as dx/dy = psi(x/y), use x = vy. This gives dx/dy = v + y(dv/dy) = psi(v).
Reducible to Homogeneous (Non-homogeneous linear type): dy/dx = (a1x + b1y + c1)/(a2x + b2y + c2)
Case 1: a1/a2 != b1/b2 — Translate: x = X+h, y = Y+k, choose h,k to make c1,c2 vanish. Case 2: a1/a2 = b1/b2 = m — Substitute v = a1x + b1y. The equation becomes separable.