# Homogeneous differential equation

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

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

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

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  $\lambda x$  and  $\lambda y$,  we find:

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

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

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

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:

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

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:

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

Thus,

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

### Solution method

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

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

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

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

thus transforming the original differential equation into the separable form:

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

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):

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

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  $\phi(x)$  is a solution, so is  $c \phi(x)$, 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:

$L(y) = 0 \,$

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

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

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

For example, the following differential equation is homogeneous

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

whereas the following two are inhomogeneous:

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