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101.63.174.28: Undid revision 584994427 by 101.63.174.28 (talk)

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|Homogeneous function}}
'''Definition'''. A function  <math>f(x)</math>  is said to be homogeneous of degree   <math>n</math>   if, by introducing a constant parameter  <math>\lambda</math>, replacing the variable   <math>x</math>   with   <math>\lambda x</math>   we find:
:<math> f(\lambda x) = \lambda^n f(x)\,. </math>

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

:<math>f(\lambda x, \lambda y) = \lambda^n f(x,y)\,. </math>

'''Example.''' The function  <math>f(x,y) = (2x^2-3y^2+4xy)</math>  is a homogeneous function of degree 2 because:
:<math>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).</math>
<br />
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}}

A first-order [[ordinary differential equation]] in the form:

:<math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>

is a homogeneous type if both functions ''M''(''x, y'') and ''N''(''x, y'') are [[homogeneous function]]s of the same degree ''n''.<ref>{{harvnb|Ince|1956|p=18}}</ref> That is, multiplying each variable by a parameter  <math>\lambda</math>, we find:

:<math>M(\lambda x, \lambda y) = \lambda^n M(x,y) </math> <span style="font-size: 1.2em;">     and     </span> <math> N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. </math>

Thus,
:<math>\frac{M(\lambda x, \lambda y)}{N(\lambda x, \lambda y)} = \frac{M(x,y)}{N(x,y)}\,. </math>

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

:<math>\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)\,. </math>

Introduce the [[change of variables]] <math>y=ux</math>; differentiate using the [[product rule]]:

:<math>\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u,</math>

thus transforming the original differential equation into the [[Separation of variables|separable]] form:
: <math>x\frac{du}{dx} = f(u) - u\,; </math>

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):
:<math> (ax + by + c) dx + (ex + fy + g) dy = 0\, , </math>

can be transformed into a homogeneous type by a linear transformation of both variables (<math>\alpha</math> and <math>\beta</math> are constants):
:<math>t = x + \alpha; \,\,\,\, z = y + \beta \,. </math>

==Homogeneous linear differential equations==

'''Definition.''' A linear differential equation is called '''homogeneous''' if the following condition is satisfied: If  <math>\phi(x)</math>  is a solution, so is  <math>c \phi(x)</math>, where <math>c</math> 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:

:<math> L(y) = 0 \,</math>
<math>
</math>where ''L'' is a [[differential operator]], a sum of derivatives, each multiplied by a function  <math>f_i</math>  of ''x'':

:<math> L = \sum_{i=1}^n f_i(x)\frac{d^i}{dx^i} \,; </math>
where  <math>f_i</math>  may be constants, but not all  <math>f_i</math>  may be zero.

For example, the following differential equation is homogeneous

:<math> \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 \,, </math>

whereas the following two are inhomogeneous:

:<math> 2 x^2 \frac{d^2y}{dx^2} + 4 x \frac{dy}{dx} + y = \cos(x) \,; </math>

:<math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>

==See also==
* [[Method of separation of variables]]

==Notes==
{{Reflist}}

==References==
* {{citation | last1=Boyce | first1=William E. | last2=DiPrima | first2=Richard C. | title = Elementary differential equations and boundary value problems | year=2012 | publisher=Wiley | isbn=978-0470458310 | edition=10th}}. (This is a good introductory reference on differential equations.)
* {{citation | last1=Ince | first1=E. L. | title=Ordinary differential equations | url=http://archive.org/details/ordinarydifferen029666mbp | year=1956 | publisher=Dover Publications | location=New York | isbn=0486603490}}. (This is a classic reference on ODEs, first published in 1926.)

==External links==
*[http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html Homogeneous differential equations at MathWorld]
*[http://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Substitution_1 Wikibooks: Ordinary Differential Equations/Substitution 1]

[[Category:Differential equations]]

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