# 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  ${\displaystyle f(x)}$  is said to be homogeneous of degree   ${\displaystyle n}$   if, by introducing a constant parameter  ${\displaystyle \lambda }$, replacing the variable   ${\displaystyle x}$   with   ${\displaystyle \lambda x}$   we find:

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

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

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

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

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

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:

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

is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n.[1] That is, multiplying each variable by a parameter  ${\displaystyle \lambda }$, we find:

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

Thus,

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

### Solution method

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

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

Introduce the change of variables ${\displaystyle y=ux}$; differentiate using the product rule:

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

thus transforming the original differential equation into the separable form:

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

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):

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

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

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

## Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If  ${\displaystyle \phi (x)}$  is a solution, so is  ${\displaystyle c\phi (x)}$, where ${\displaystyle c}$ is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y; a constant term breaks homogeneity. 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:

${\displaystyle L(y)=0\,}$

where L is a differential operator, a sum of derivatives, each multiplied by a function  ${\displaystyle f_{i}}$  of x:

${\displaystyle L=\sum _{i=1}^{n}f_{i}(x){\frac {d^{i}}{dx^{i}}}\,;}$

where  ${\displaystyle f_{i}}$  may be constants, but not all  ${\displaystyle f_{i}}$  may be zero.

For example, the following differential equation is homogeneous

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

whereas the following two are inhomogeneous:

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