Homogeneous differential equations

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

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

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 \,.

See also

References

  • Elementary Differential Equations and Boundary Value Problems (10th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 2012, ISBN 978-0470458310. (This is a good introductory reference on differential equations.)
  • Ordinary Differential Equations, Ince, E. L., available as paperback from Dover publication and also as a free download from Internet Archive http://archive.org/. (This is a classic reference on ODEs.)

External links


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