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.

Homogeneous functions

Main article: Homogeneous function

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

Differential equations

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

Scope

Natural sciences Engineering
Astronomy Physics Chemistry Biology Geology
Applied mathematics
Continuum mechanics Chaos theory Dynamical systems
Social sciences
Economics Population dynamics

Classification

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

Relation to processes

Difference (discrete analogue)

Solution

General topics

Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions

Solution methods

People

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.^[1] 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).

The equations in this discussion are not to be used as formulary for solutions; they are shown just to demonstrate the method of solution.

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. 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:

L(y) = 0 \,

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 $f_i$ of x:

L = \sum_{i=0}^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 \,.

Note: the existence of a constant term is enough for this equation to be inhomogeneous.

Notes

↑ Ince 1956, p. 18

References

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.)