### Homogeneous differential equations

The term "homogeneous" is used in more than one context in mathematics. Perhaps the most prominent are the following three distinct cases:

- Homogeneous functions.
- Homogeneous type of first order differential equations.
- Homogeneous differential equations (in contrast to "inhomogeneous" differential equations). This definition is used to define a property of certain linear differential equations—it is unrelated to the above two cases.

Each one of these cases will be briefly explained as follows.

## Contents

## Homogeneous Functions

**Definition**. A function $f(x)$ is said to be homogeneous of degree $n$ if, by introducing a constant parameter $\backslash lambda$, replacing the variable $x$ with $\backslash lambda\; x$ we find:

- $f(\backslash lambda\; x)\; =\; \backslash lambda^n\; f(x)\backslash ,.$

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables $f(x,y)$ is said to be homogeneous of degree $n$ if we replace both variables $x$ and $y$ by $\backslash lambda\; x$ and $\backslash lambda\; y$, we find:

- $f(\backslash lambda\; x,\; \backslash lambda\; y)\; =\; \backslash lambda^n\; f(x,y)\backslash ,.$

**Example.** The function $f(x,y)\; =\; (2x^2-3y^2+4xy)$ is a homogeneous function of degree 2 because:

- $f(\backslash lambda\; x,\; \backslash lambda\; y)\; =\; [2(\backslash lambda\; x)^2-3(\backslash lambda\; y)^2+4(\backslash lambda\; x\; \backslash lambda\; y)]\; =\; (2\backslash lambda^2x^2-3\backslash lambda^2y^2+4\backslash lambda^2\; xy)\; =\; \backslash lambda^2(2x^2-3y^2+4xy)=\backslash lambda^2f(x,y).$

This definition of homogeneous functions has been used to classify certain types of first order differential equations.

## Homogeneous Type of First Order Differential Equations

A first-order ordinary differential equation in the form:

- $M(x,y)\backslash ,dx\; +\; N(x,y)\backslash ,dy\; =\; 0$

is a homogeneous type if both functions *M*(*x, y*) and *N*(*x, y*) are homogeneous functions of the same degree *n.* That is, multiplying each variable by a parameter $\backslash lambda$, we find:

- $M(\backslash lambda\; x,\; \backslash lambda\; y)\; =\; \backslash lambda^n\; M(x,y)$ and $N(\backslash lambda\; x,\; \backslash lambda\; y)\; =\; \backslash lambda^n\; N(x,y)\backslash ,.$

Thus,

- $\backslash frac\{M(\backslash lambda\; x,\; \backslash lambda\; y)\}\{N(\backslash lambda\; x,\; \backslash lambda\; y)\}\; =\; \backslash frac\{M(x,y)\}\{N(x,y)\}\backslash ,.$

### Solution method

In the quotient $\backslash frac\{M(tx,ty)\}\{N(tx,ty)\}\; =\; \backslash frac\{M(x,y)\}\{N(x,y)\}$, we can let $t\; =\; 1/x$ to simplify this quotient to a function $f$ of the single variable $y/x$:

- $\backslash frac\{M(x,y)\}\{N(x,y)\}\; =\; \backslash frac\{M(tx,ty)\}\{N(tx,ty)\}\; =\; \backslash frac\{M(1,y/x)\}\{N(1,y/x)\}=f(y/x)\backslash ,.$

Introduce the change of variables $y=ux$; differentiate using the product rule:

- $\backslash frac\{d(ux)\}\{dx\}\; =\; x\backslash frac\{du\}\{dx\}\; +\; u\backslash frac\{dx\}\{dx\}\; =\; x\backslash frac\{du\}\{dx\}\; +\; u,$

thus transforming the original differential equation into the separable form:

- $x\backslash frac\{du\}\{dx\}\; =\; f(u)\; -\; u\backslash ,;$

this form can now be integrated directly (see ordinary differential equation).

### Special Case

A first order differential equation of the form (*a*, *b*, *c*, *e*, *f*, *g* are all constants):

- $(ax\; +\; by\; +\; c)\; dx\; +\; (ex\; +\; fy\; +\; g)\; dy\; =\; 0\backslash ,\; ,$

can be transformed into a homogeneous type by a linear transformation of both variables ($\backslash alpha$ and $\backslash beta$ are constants):

- $t\; =\; x\; +\; \backslash alpha;\; \backslash ,\backslash ,\backslash ,\backslash ,\; z\; =\; y\; +\; \backslash beta\; \backslash ,.$

## Homogeneous linear differential equations

**Definition.** A linear differential equation is called **homogeneous** if the following condition is satisfied: If $\backslash phi(x)$ is a solution, so is $c\; \backslash phi(x)$, where $c$ is an arbitrary (non-zero) constant. A linear differential equation that fails this condition is called **inhomogeneous.**

A linear differential equation can be represented as a linear operator acting on *y(x)* where *x* is usually the independent variable and *y* is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is of the form:

- $L(y)\; =\; 0\; \backslash ,$

$$where *L* is a differential operator, a sum of derivatives, each multiplied by a functions $f\_i$ of *x*:

- $L\; =\; \backslash sum\_\{i=1\}^n\; f\_i(x)\backslash frac\{d^i\}\{dx^i\}\; \backslash ,;$

where $f\_i$ may be constants, but not all $f\_i$ may be zero.

For example, the following differential equation is homogeneous

- $\backslash sin(x)\; \backslash frac\{d^2y\}\{dx^2\}\; +\; 4\; \backslash frac\{dy\}\{dx\}\; +\; y\; =\; 0\; \backslash ,,$

whereas the following two are inhomogeneous:

- $2\; x^2\; \backslash frac\{d^2y\}\{dx^2\}\; +\; 4\; x\; \backslash frac\{dy\}\{dx\}\; +\; y\; =\; \backslash cos(x)\; \backslash ,;$

- $2\; x^2\; \backslash frac\{d^2y\}\{dx^2\}\; -\; 3\; x\; \backslash frac\{dy\}\{dx\}\; +\; y\; =\; 2\; \backslash ,.$

## See also

## References

- Elementary Differential Equations and Boundary Value Problems (10th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 2012, ISBN 978-0470458310. (This is a good introductory reference on differential equations.)
- Ordinary Differential Equations, Ince, E. L., available as paperback from Dover publication and also as a free download from Internet Archive http://archive.org/. (This is a classic reference on ODEs.)

## External links

- Homogeneous differential equations at MathWorld
- : Ordinary Differential Equations/Substitution 1

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