View source for Homogeneous differential equation

A [[differential equation]] can be '''homogeneous''' in either of two respects. 

A [[first order differential equation]] is said to be homogeneous if it may be written
:<math>f(x,y)dy = g(x,y)dx,</math>
where {{mvar|f}} and {{mvar|g}} are [[homogeneous function]]s of the same degree of {{mvar|x}} and {{mvar|y}}.<ref name="Zill2012">{{cite book|author=Dennis G. Zill|title=A First Course in Differential Equations with Modeling Applications|url=https://books.google.com/books?id=pasKAAAAQBAJ&printsec=frontcover#v=onepage&q=homogeneous&f=false|date=15 March 2012|publisher=Cengage Learning|isbn=1-285-40110-7}}</ref> In this case, the change of variable {{math|1=''y'' = ''ux''}} leads to an equation of the form 
:<math>\frac{dx}x = h(u)du,</math>
which is easy to solve by [[integral|integration]] of the two members.

Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of [[linear differential equation]]s, this means that there are no constant terms. The solutions of any linear [[ordinary differential equation]] of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.

== Homogeneous 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 &nbsp;<math>\lambda</math>, we find

:<math>M(\lambda x, \lambda y) = \lambda^n M(x,y)\,</math> <span style="font-size: 1.2em;"> &nbsp; &nbsp; and &nbsp; &nbsp; </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 &nbsp; <math>\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>,
we can let &nbsp; <math>t = 1/x</math> &nbsp; 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>
That is 
:<math>\frac{dy}{dx}=-f(y/x).</math>

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

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

This transforms the original differential equation into the [[Separation of variables|separable]] form
: <math>x\frac{du}{dx} = -f(u) - u, </math>
or
: <math>\frac 1x\frac{dx}{du} = \frac {-1}{f(u) + u}, </math>
which can now be integrated directly: {{math|log ''x''}} equals the [[antiderivative]] of the right-hand side (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>
where ''af'' ≠ ''be''
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==
{{see also|Linear differential equation}}
A linear differential equation is  '''homogeneous''' if it is a [[homogeneous linear equation]] in the unknown function and its derivatives. It follows that, if <math>\phi(x)</math> is a solution, so is <math>c \phi(x)</math>, for any (non-zero) constant {{mvar|c}}. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. 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 

:<math> L(y) = 0 \,</math>
<math>
</math>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 &nbsp;<math>f_i</math>&nbsp; of ''x'':

:<math> L = \sum_{i=0}^n f_i(x)\frac{d^i}{dx^i} \, ,</math>
where &nbsp;<math>f_i</math>&nbsp; may be constants, but not all &nbsp;<math>f_i</math>&nbsp; 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>
The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.

==See also==
* [[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=https://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.)
* {{cite book|author1=Andrei D. Polyanin|author2=Valentin F. Zaitsev|title=Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems|url=https://books.google.com/books?id=L3JQDwAAQBAJ&printsec=frontcover#v=onepage&q=homogeneous&f=false|date=15 November 2017|publisher=CRC Press|isbn=978-1-4665-6940-9}}
* {{cite book|author1=Matthew R. Boelkins|author2=Jack L. Goldberg|author3=Merle C. Potter|title=Differential Equations with Linear Algebra|url=https://books.google.com/books?id=s1-mZMg5fLgC&pg=PA274&dq=homogeneous+differential+equations&hl=en&sa=X&ved=0ahUKEwjStIGfrMHjAhW3B50JHeDkANgQ6AEIWDAJ#v=onepage&q=homogeneous&f=false|date=5 November 2009|publisher=Oxford University Press|isbn=978-0-19-973666-9|pages=274–}}

==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]]
[[Category:Ordinary differential equations]]