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:
- 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.
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.
Homogeneous type of first-order differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction.
A first-order ordinary differential equation in the form:
In the quotient , we can let to simplify this quotient to a function of the single variable :
thus transforming the original differential equation into the separable form:
this form can now be integrated directly (see ordinary differential equation).
The equations in this discussion are not to be used as formulary for solutions; they are shown just to demonstrate the method of solution.
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):
Homogeneous linear differential equations
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 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 (defining the "0th derivative" as the original, non-differentiated function), 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:
Note: the existence of a constant term is enough for this equation to be inhomogeneous.
- Ince 1956, p. 18
- 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.)