Homogeneous Form or Homogeneous Differential Equation

Definition 10.12 (Homogeneous function of degree $n$). A function $f(x,y)$ is homogeneous of degree $n$ if $f(tx,ty)=t^n f(x,y)$ for all suitably restricted $x,y,t$ (this scaling property is called Euler's homogeneity). For example, $f(x,y)=6x^2+2xy+4y^2$ is homogeneous of degree $2$ (every term has total degree $2$ in $x,y$); but $f(x,y)=x^3+(\sin x)e^y$ is not homogeneous, since $\sin x$ and $e^y$ do not scale as pure powers of $t$.

If $f(x,y)$ is homogeneous of degree zero, it can always be written purely in terms of the ratio $\dfrac{y}{x}$ or $\dfrac{x}{y}$: $f(x,y)=g\!\left(\dfrac{y}{x}\right)$ (or $g\!\left(\dfrac{x}{y}\right)$).

Definition 10.13 (Homogeneous Differential Equation). An ODE is in homogeneous form if it is written as $\dfrac{dy}{dx}=g\!\left(\dfrac{y}{x}\right)$.

Caution: this use of the word "homogeneous" (Definition 10.13, applied to the differential equation) is a different meaning from Definition 10.7's "homogeneous" (applied to a linear equation's right side being zero) — the same word is used for two distinct ideas in this chapter.

Remark. The differential form $M(x,y)\,dx+N(x,y)\,dy=0$ is homogeneous exactly when $M$ and $N$ are homogeneous functions of the same degree; equivalently, writing it as $\dfrac{dy}{dx}=f(x,y)$ with $f(x,y)=-M(x,y)/N(x,y)$, $f$ is automatically homogeneous of degree $0$. For instance, $\left(x^2-3y^2\right)dx+2xy\,dy=0$ rewrites as $\dfrac{dy}{dx}=\dfrac{3y}{2x}-\dfrac{x}{2y}=g\!\left(\dfrac{y}{x}\right)$ — homogeneous. But $\dfrac{dy}{dx}=\dfrac{x^3+y^2x}{x^3-xy^2}$ is not homogeneous (the given right side does not reduce to a pure function of $y/x$).

Theorem 10.1 (Solution method). If $M(x,y)\,dx+N(x,y)\,dy=0$ is homogeneous, the substitution $y=vx$ transforms it into a separable equation in $v$ and $x$. Concretely: with $y=vx$, $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$; substituting into $\dfrac{dy}{dx}=g\!\left(\dfrac{y}{x}\right)=g(v)$ gives

which is separable: $\dfrac{dv}{g(v)-v}=\dfrac{dx}{x}$. Integrate, then replace $v$ by $\dfrac{y}{x}$.

When to use $x=vy$ instead. If the natural ratio in the equation is $\dfrac{x}{y}$ (e.g. an equation more cleanly written as $\dfrac{dx}{dy}=g\!\left(\dfrac{x}{y}\right)$), substitute $x=vy$ instead, giving $\dfrac{dx}{dy}=v+y\dfrac{dv}{dy}$, and separate in $v,y$ the same way.

Worked examples illustrate the range of homogeneous equations:

• $\left(x^2-3y^2\right)dx+2xy\,dy=0$: substitution $y=vx$ leads to $\dfrac{2v\,dv}{v^2-1}=\dfrac{dx}{x}$, integrating to the algebraic solution $y^2-x^2=kx^3$.
• $\left(y+\sqrt{x^2+y^2}\right)dx-x\,dy=0,\ y(1)=0$: with $y=vx$, separates to a logarithmic solution; the initial condition fixes the constant, giving the particular solution $\sqrt{x^2+y^2}+y=x^2$. …