Homogeneous differential equation

A homogeneous differential equation has several distinct meanings.

One meaning is that a first-order ordinary differential equation is homogeneous if it has the form : $frac\{dy\}\{dx\}\; =\; F(y/x).$To solve such equations, one makes the change of variables "u" = "y"/"x", which will transform such an equation into separable one.

Another meaning is a linear homogeneous differential equation, which is a differential equation of the form

: $Ly\; =\; 0\; ,$

where the differential operator "L" is a linear operator, and "y" is the unknown function.

Example of deriving a homogenous equation

A well known homogenous equation in x and y of degree m, subsequently showing one of Euler's identities is as follows.

: $f(x,y)\; =\; x^m\; Fleft(frac\{y\}\{x\}\; ight).$Deriving $f\_x\; (x,y)$We obtain the following,

$frac\{partial\; f(x,y)\}\{partial\; x\}\; =\; mx^\{m-1\}Fleft(frac\{y\}\{x\}\; ight)+\; x^mF^\text{'}left(frac\{y\}\{x\}\; ight)cdotleft(-frac\{y\}\{x^2\}\; ight)$.

Where $F^\text{'}$ denotes the first derivative of F with respect to the homogenous argument.

Also,

$frac\{partial\; f(x,y)\}\{partial\; y\}\; =\; x^mF^\text{'}left(frac\{y\}\{x\}\; ight).left(frac\{1\}\{x\}\; ight).$

Now taking each derivative and multiplying by its corresponding variable we arrive at the following equation.

$xfrac\{partial\; f(x,y)\}\{partial\; x\}\; +\; yfrac\{partial\; f(x,y)\}\{partial\; y\}\; =\; xleft\; [mx^\{m-1\}Fleft(frac\{y\}\{x\}\; ight)+\; x^m.left(-frac\{y\}\{x^2\}\; ight)F^\text{'}left(frac\{y\}\{x\}\; ight)\; ight]\; +\; yleft\; [\; x^m.left(frac\{1\}\{x\}\; ight)F^\text{'}left(frac\{y\}\{x\}\; ight)\; ight]$

$xfrac\{partial\; f(x,y)\}\{partial\; x\}\; +\; yfrac\{partial\; f(x,y)\}\{partial\; y\}\; =\; xleft\; [mx^\{m-1\}Fleft(frac\{y\}\{x\}\; ight)-x^\{m-2\}yF^\text{'}left(frac\{y\}\{x\}\; ight)\; ight]\; +\; yleft\; [\; x^\{m-1\}F^\text{'}left(frac\{y\}\{x\}\; ight)\; ight]$

: $=\; mx^mFleft(frac\{y\}\{x\}\; ight)$

: $=\; mf(x,y).$

Which in turn is one of Euler's identities,

$xfrac\{partial\; f(x,y)\}\{partial\; x\}\; +\; yfrac\{partial\; f(x,y)\}\{partial\; y\}\; =\; mf(x,y)$

This identity is generalized by Euler's theorem on homogeneous functions.

External links

* [http://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html Homogeneous differential equations at MathWorld]
* [http://math.stcc.edu/DiffEq/DiffEQ41.html Homogeneous Differential equations]
* [http://en.wikibooks.org/wiki/Differential_Equations/Substitution_1 Wikibooks: Differential Equations/First-Order/Substitution Methods]