Can a differential equation be non-linear and homogeneous at the same time?

Question

I have searched for the definition of homogeneous differential equation. I have found definitions of linear homogeneous differential equation. Can a differential equation be non-linear and homogeneous at the same time? (If yes then) what is the definition of homogeneous differential equation in general? y'' + sin(y) = 0 is it homogeneous?

Answer 1

Well In my book its given that any function "$f(x,y)$" satisfying "$f(\lambda x,\lambda y) = \lambda ^n f(x,y) $" where $n$ is any integer, is a homogeneous function and differential equation which involve homogeneous function is called homogeneous differential equation.

Well for the question if a non-linear differential equation can be homogeneous or not. Yes, of course it can be. Consider the differential equation,

$\frac{\mathrm dy}{\mathrm d x} = \frac{y^2-xy+x^2sin(\frac{y}{x})}{x^2} $ .

This equation is neither linear in x or y but it is homogeneous. As,

$f(x,y)=\frac{y^2-xy+x^2\sin (\frac {y}{x})}{x^2}$

$f(\lambda x,\lambda y)=\frac{\lambda ^2y^2-\lambda ^2xy+\lambda ^2x^2\sin (\frac {\lambda y}{\lambda x})}{\lambda ^2x^2}$

Now dividing numerator and denominator by $\lambda ^2$

$f(\lambda x,\lambda y)=\frac{y^2-xy+x^2\sin (\frac {y}{x})}{x^2} = \lambda ^0 .f(x,y)$

Hence the function and so the differential equation is homogeneous. Here neither x or y is linear but the differential equation is homogeneous.

Answer 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.