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| The term "''''''homogeneous''''''" is used in more than one context in mathematics. Perhaps the most prominent are the following three distinct cases:
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| # Homogeneous functions
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| # Homogeneous type of first order differential equations
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| # 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.
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| Each one of these cases will be briefly explained as follows.
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| == Homogeneous functions ==
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| {{main|Homogeneous function}}
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| '''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>
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| 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:
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| :<math>f(\lambda x, \lambda y) = \lambda^n f(x,y)\,. </math>
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| '''Example.''' The function <math>f(x,y) = (2x^2-3y^2+4xy)</math> is a homogeneous function of degree 2 because:
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| :<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>
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| <br />
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| This definition of homogeneous functions has been used to classify certain types of first order differential equations.
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| == Homogeneous type of first-order differential equations ==
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| {{Differential equations}}
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| A first-order [[ordinary differential equation]] in the form:
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| :<math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>
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| 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: | |
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| :<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>
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| Thus,
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| :<math>\frac{M(\lambda x, \lambda y)}{N(\lambda x, \lambda y)} = \frac{M(x,y)}{N(x,y)}\,. </math>
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| ===Solution method===
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| In the quotient <math>\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>,
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| 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>:
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| :<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>
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| Introduce the [[change of variables]] <math>y=ux</math>; differentiate using the [[product rule]]:
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| :<math>\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u,</math>
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| thus transforming the original differential equation into the [[Separation of variables|separable]] form:
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| : <math>x\frac{du}{dx} = f(u) - u\,; </math>
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| this form can now be integrated directly (see [[ordinary differential equation]]).
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| ===Special case===
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| A first order differential equation of the form (''a'', ''b'', ''c'', ''e'', ''f'', ''g'' are all constants):
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| :<math> (ax + by + c) dx + (ex + fy + g) dy = 0\, , </math>
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| can be transformed into a homogeneous type by a linear transformation of both variables (<math>\alpha</math> and <math>\beta</math> are constants):
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| :<math>t = x + \alpha; \,\,\,\, z = y + \beta \,. </math>
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| ==Homogeneous linear differential equations==
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| '''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.'''
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| 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:
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| :<math> L(y) = 0 \,</math>
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| <math>
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| </math>where ''L'' is a [[differential operator]], a sum of derivatives, each multiplied by a function <math>f_i</math> of ''x'':
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| :<math> L = \sum_{i=1}^n f_i(x)\frac{d^i}{dx^i} \,; </math>
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| where <math>f_i</math> may be constants, but not all <math>f_i</math> may be zero.
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| For example, the following differential equation is homogeneous
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| :<math> \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 \,, </math>
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| whereas the following two are inhomogeneous:
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| :<math> 2 x^2 \frac{d^2y}{dx^2} + 4 x \frac{dy}{dx} + y = \cos(x) \,; </math>
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| :<math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>
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| ==See also==
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| * [[Method of separation of variables]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{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.)
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| * {{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.)
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| ==External links==
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| *[http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html Homogeneous differential equations at MathWorld]
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| *[http://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Substitution_1 Wikibooks: Ordinary Differential Equations/Substitution 1]
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| [[Category:Differential equations]]
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