What is a homogeneous Differential Equation?

In first-order ODEs, we say that a differential equation in the form $$\frac{\mathrm d y}{\mathrm d x}=f(x,y)$$ is said to be homogeneous if the function $f(x,y)$ can be expressed in the form $f\left(\displaystyle\frac{y}{x}\right)$, and then solved by the substitution $z=\displaystyle\frac{y}{x}$.

In second-order ODEs, we say that a differential equation in the form $$a\frac{\mathrm d^2 y}{\mathrm d x^2}+b\frac{\mathrm d y}{\mathrm d x}+cy=f(x)$$ is said to be homogeneous if $f(x)=0$.

Is there a relation between these two? What does homogeneous mean? I thought it's when something $=0$, because in linear algebra, a system of $n$ equations is homogeneous if it is in the form $\boldsymbol{\mathrm{Ax}}=\boldsymbol 0_{n\times 1}$; but this doesn't seem to be the case for first-order ODEs.

Solution 1:

The term homogeneity refers to a scaling property: A function $f$ is homogeneous if $f(\lambda x) = \lambda^\alpha f(x)$ for all $\lambda > 0$ and some real number $\alpha$ (the degree).

For the first case, observe that if $f$ is homogeneous of degree 0 in both variables, ie. $f(\lambda x, \lambda y) = f(x,y)$, then it can be expressed as $f(x,y) = g(y/x)$.

A linear differential equation $Ly = f$ with $f = 0$ is called homogeneous, because if $y$ is a solution of $Ly = 0$ then $\lambda y$ also solves the equation.

Solution 2:

General Homogeneity

Ibragimov A Practical Course in Differential Equations and Mathematical Modeling, §3.1.3 "Homogeneous Equations", p. 93:

An ordinary differential equation of an arbitrary order $$F(x,y,y',…,y^{(n)})=0$$ is said to be homogeneous [in general] if it is invariant under a scaling transformation (dilation) of the independent and dependent variables […]: $$\bar{x}=a^kx,\qquad\bar{y}=a^ly,$$ where $a>0$ is a parameter not identical with 1, and $k$ and $l$ are any fixed real numbers. The invariance means that $$F(\bar{x},\bar{y},\bar{y}',…,\bar{y}^{(n)})=0,$$ where $\bar{y}'=d\bar{y}/d\bar{x}$, etc.

Double Homogeneity

Ibid. p. 95:

[A differential equation] is double homogeneous if […] it does not alter under the transformations $$\bar{x}=ax,\qquad\bar{y}=y,$$ and $$\bar{x}=x,\qquad\bar{y}=ay,$$ with independent positive parameters $a$ and $b$, respectively.

Type 1: Uniform Homogeneity

Ibid. §3.1.4 "Different types of homogeneity", p. 96:

The uniformly homogeneous equations are invariant under the uniform scaling: $$\bar{x}=ax,\qquad \bar{y}=ay$$

The general form of this type of homogeneous equations is (cf. MathWorld): $$\frac{dy}{dx}=F\left(\frac{y}{x}\right).$$

Type 2: Homogeneity by Function

Ibid. p. 97:

This type of homogeneity designates invariance […] with respect to the dilation of $y$ only: $$\bar{x}=x,\qquad \bar{y}=ay$$

This is the more common understanding of homogeneity:

Licker's Dictionary of Mathematics p. 108 defines a homogeneous differential equation as

A differential equation where every scalar multiple of a solution is also a solution.

Zwillinger's Handbook of Differential Equations p. 6:

An equation is said to be homogeneous if all terms depend linearly on the dependent variable or its derivatives. For example, the equation $y_{xx} + xy = 0$ is homogeneous while the equation $y_{xx} + y = 1$ is not.

Olver's Introduction to Partial Differential Equations p. 9:

A differential equation is called homogeneous linear if both sides are sums of terms, each of which involves the dependent variable $u$ or one of its derivatives to the first power; on the other hand, there is no restriction on how the terms involve the independent variables. Thus, $$\frac{d^2u}{dx^2}+\frac{u}{1+x^2}=0$$ is a homogeneous linear second-order ordinary differential equation.