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?

  • 2
    $\begingroup$ No, the word is only used in the context of linear equations. $\endgroup$ Jul 16 '15 at 18:57
  • 2
    $\begingroup$ It could be said homogeneous if it is expressed in the form:$p(y',y)=0$ where $p$ is a homogeneous polynomial of two variables. Homogeneity is a notion related to polynomials of several variables. $\endgroup$
    – Bernard
    Jul 16 '15 at 19:07
  • 2
    $\begingroup$ @DavidC.Ullrich Actually, differential equations of form $y' = f(\frac{y}{x})$ are also called homogeneous. To OP: I think you've missed this entry in Wikipedia: en.wikipedia.org/wiki/Homogeneous_differential_equation $\endgroup$
    – Evgeny
    Jul 16 '15 at 19:25
  • $\begingroup$ @Evgeny Thanx for reply. I re-read that wikipedia page, but didn't get my answer. $\endgroup$
    – Siddhartha
    Jul 16 '15 at 20:18

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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.