en>GoodDay at 10:27, 2 July 2014

2014-07-02T10:27:59Z

101.63.174.28: Undid revision 584994427 by 101.63.174.28 (talk)

2013-12-07T14:28:11Z

Undid revision 584994427 by 101.63.174.28 (talk)

← Older revision		Revision as of 16:28, 7 December 2013
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	The ~~author~~ is ~~known~~ by the ~~title~~ of ~~Numbers Lint~~. The ~~preferred hobby for my kids~~ and me is to ~~play baseball but I haven~~'t ~~made~~ a ~~dime with it. California~~ is ~~exactly~~ where ~~her home~~ is ~~but she requirements~~ to ~~transfer because~~ of ~~her family~~. ~~Hiring~~ is ~~his profession~~.<br><br>~~Here~~ is ~~my blog post~~ [http://~~www~~.~~youporntime~~.com/~~user~~/~~KBiaggini www~~.~~youporntime~~.~~com~~]		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]]

en>Christian75: clean up, typos fixed: , → , using AWB (7916)

2012-01-27T22:42:22Z

clean up, typos fixed: , → , using AWB (7916)

New page

The author is known by the title of Numbers Lint. The preferred hobby for my kids and me is to play baseball but I haven't made a dime with it. California is exactly where her home is but she requirements to transfer because of her family. Hiring is his profession.<br><br>Here is my blog post [http://www.youporntime.com/user/KBiaggini www.youporntime.com]

Band emission - Revision history

en>GoodDay at 10:27, 2 July 2014

101.63.174.28: Undid revision 584994427 by 101.63.174.28 (talk)

en>Christian75: clean up, typos fixed: , → , using AWB (7916)