Homogeneous Differential Equations

A careful treatment of first-order homogeneous equations $y'=F(y/x)$ and the substitution $v=y/x$.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

A first-order equation is homogeneous in the substitution sense when it can be written $y'=F(y/x)$, or when $M(x,y)dx+N(x,y)dy=0$ has $M,N$ homogeneous of the same degree. Setting $v=y/x$ so $y=vx$ and $y'=v+xv'$ converts it to a separable equation.

Notation and mathematical language

This meaning differs from a homogeneous linear equation, whose forcing is zero. Here $v(x)=y(x)/x$, so the method works on intervals with $x\ne0$. After substitution, $xv'=F(v)-v$ and, where the denominator is nonzero, $dv/[F(v)-v]=dx/x$.

Conceptual picture

The equation is unchanged along rays from the origin because $y/x$ is constant on each ray. The substitution tracks the evolving ray slope $v$. Roots of $F(v)-v=0$ give straight-line solutions $y=vx$ and can be lost during separation.

Fully worked example

Interpretation and application

These equations model scale-invariant slopes and geometric families where only a ratio matters. Scale invariance is a property of the mathematical formulation; measurement offsets or fixed external scales can break it and require another model.