From Wikipedia, the free encyclopedia - View original article

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.

Main article: Homogeneous function

**Definition**. A function is said to be homogeneous of degree if, by introducing a constant parameter , replacing the variable with we find:

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables is said to be homogeneous of degree if we replace both variables and by and , we find:

**Example.** The function is a homogeneous function of degree 2 because:

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

Differential equations |
---|

Navier–Stokes differential equations used to simulate airflow around an obstruction. |

Classification |

Types |

Relation to processes |

Solution |

General topics |

A first-order ordinary differential equation in the form:

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 , we find:

- and

Thus,

In the quotient , we can let to simplify this quotient to a function of the single variable :

Introduce the change of variables ; differentiate using the product rule:

thus transforming the original differential equation into the separable form:

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

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

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

**Definition.** A linear differential equation is called **homogeneous** if the following condition is satisfied: If is a solution, so is , where 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:

where *L* is a differential operator, a sum of derivatives, each multiplied by a function of *x*:

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

For example, the following differential equation is homogeneous

whereas the following two are inhomogeneous:

- Boyce, William E.; DiPrima, Richard C. (2012),
*Elementary differential equations and boundary value problems*(10th ed.), Wiley, ISBN 978-0470458310. (This is a good introductory reference on differential equations.) - Ince, E. L. (1956),
*Ordinary differential equations*, New York: Dover Publications, ISBN 0486603490. (This is a classic reference on ODEs, first published in 1926.)