### 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:

1. Homogeneous functions.
2. Homogeneous type of first order differential equations.
3. 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.

## Homogeneous Functions

Main article: Homogeneous function

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

$f\left(\lambda x\right) = \lambda^n f\left(x\right)\,.$

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

$f\left(\lambda x, \lambda y\right) = \lambda^n f\left(x,y\right)\,.$

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

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

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:

$M\left(x,y\right)\,dx + N\left(x,y\right)\,dy = 0$

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  $\lambda$, we find:

$M\left(\lambda x, \lambda y\right) = \lambda^n M\left(x,y\right)$     and     $N\left(\lambda x, \lambda y\right) = \lambda^n N\left(x,y\right)\,.$

Thus,

$\frac\left\{M\left(\lambda x, \lambda y\right)\right\}\left\{N\left(\lambda x, \lambda y\right)\right\} = \frac\left\{M\left(x,y\right)\right\}\left\{N\left(x,y\right)\right\}\,.$

### Solution method

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

$\frac\left\{M\left(x,y\right)\right\}\left\{N\left(x,y\right)\right\} = \frac\left\{M\left(tx,ty\right)\right\}\left\{N\left(tx,ty\right)\right\} = \frac\left\{M\left(1,y/x\right)\right\}\left\{N\left(1,y/x\right)\right\}=f\left(y/x\right)\,.$

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

$\frac\left\{d\left(ux\right)\right\}\left\{dx\right\} = x\frac\left\{du\right\}\left\{dx\right\} + u\frac\left\{dx\right\}\left\{dx\right\} = x\frac\left\{du\right\}\left\{dx\right\} + u,$

thus transforming the original differential equation into the separable form:

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

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):

$\left(ax + by + c\right) dx + \left(ex + fy + g\right) dy = 0\, ,$

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

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

## Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If  $\phi\left(x\right)$  is a solution, so is  $c \phi\left(x\right)$, 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:

$L\left(y\right) = 0 \,$

where L is a differential operator, a sum of derivatives, each multiplied by a functions  $f_i$  of x:

$L = \sum_\left\{i=1\right\}^n f_i\left(x\right)\frac\left\{d^i\right\}\left\{dx^i\right\} \,;$

where  $f_i$  may be constants, but not all  $f_i$  may be zero.

For example, the following differential equation is homogeneous

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

whereas the following two are inhomogeneous:

$2 x^2 \frac\left\{d^2y\right\}\left\{dx^2\right\} + 4 x \frac\left\{dy\right\}\left\{dx\right\} + y = \cos\left(x\right) \,;$
$2 x^2 \frac\left\{d^2y\right\}\left\{dx^2\right\} - 3 x \frac\left\{dy\right\}\left\{dx\right\} + y = 2 \,.$