Homogeneous differential equations

A function F(x, y) is homogeneous of degree n if F(λx, λy) = λⁿF(x, y) for every nonzero λ. For example, x² + xy is homogeneous of degree 2, and (x² + y²)/(xy) is homogeneous of degree 0.

The equation dy/dx = F(x, y) is homogeneous when F is homogeneous of degree 0. Then F can be written as g(y/x).

Method: put y = vx, so dy/dx = v + x dv/dx. The equation becomes v + x dv/dx = g(v), which separates into dv/(g(v) − v) = dx/x. Integrate, then replace v by y/x.

Worked example: dy/dx = (x + y)/x = 1 + v. Then x dv/dx = 1, v = log|x| + C and y = x log|x| + Cx. Through (1, 2): C = 2, so y = x log x + 2x.

Worked example: dy/dx = (x² + y²)/(2xy). With y = vx, x dv/dx = (1 − v²)/(2v), so 2v dv/(1 − v²) = dx/x. Integrating gives x(1 − v²) = C, that is x² − y² = Cx.

Worked example: dy/dx = y/x + tan(y/x). Then x dv/dx = tan v, so cot v dv = dx/x and sin(y/x) = Cx.

If the equation is easier as dx/dy = G(x, y) with G of degree 0, use x = vy instead.