Determinación del grupo de simetrías de una ecuación diferencial ordinaria de primer orden
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Date
2016
Authors
Villoslada Chilón, Alexander Manuel
Journal Title
Journal ISSN
Volume Title
Publisher
Universidad Nacional de Trujillo
Abstract
En el presente trabajo el problema principal es la determinaci on de los grupos de_x000D_
simetr as de una ecuaci on diferencial de primer orden, para ello estableceremos en_x000D_
forma general la prolongaci on de un campo vectorial de nido en un subconjunto del_x000D_
plano R2, esto es, dado un campo vectorial v = (x; u) @_x000D_
@x + (x; u) @_x000D_
@u se encuentra_x000D_
la prolongaci on_x000D_
pr(n)v = _x000D_
@_x000D_
@x_x000D_
+ _x000D_
@_x000D_
@u_x000D_
+_x000D_
nX_x000D_
(k=1)_x000D_
k(x; u(n))_x000D_
@_x000D_
@u(k) ;_x000D_
para n = 1, obtenemos:_x000D_
pr(1)v = _x000D_
@_x000D_
@x_x000D_
+ _x000D_
@_x000D_
@u_x000D_
+ [ x + u0( u x) (u0)2 u]_x000D_
@_x000D_
@u0_x000D_
;_x000D_
luego le aplicamos a F(x; u; u0) e igualamos a cero (esto se convierte en una ecuacion_x000D_
diferencial parcial, llamada condici on de simetr a)._x000D_
Enseguida, conociendo el campo vectorial asociado a la ecuaci on diferencial dada,_x000D_
encontramos sus grupos de simetr as (uniparam etrico)_x000D_
x = (t; x) = etvx =_x000D_
1X_x000D_
k=0_x000D_
tk_x000D_
k!_x000D_
vkx:_x000D_
Finalmente, aplicamos la condici on de simetr a sobre ecuaciones diferenciales de_x000D_
variable separable, ecuaciones homog eneas, ecuaciones diferenciales exactas, ecuaciones diferenciales lineales, ecuaciones diferenciales no lineales (de Bernoulli y de_x000D_
Ricatti)
Description
In this paper the main problem is the determination of symmetries of a di erential_x000D_
equation of the rst order to for it establish to in generally the prolongation_x000D_
of a vector eld de ned on a subset of the plane R2, that is, given a vector eld_x000D_
v = (x; u) @_x000D_
@x + (x; u) @_x000D_
@u is the prolongation_x000D_
pr(n)v = _x000D_
@_x000D_
@x_x000D_
+ _x000D_
@_x000D_
@u_x000D_
+_x000D_
nX_x000D_
(k=1)_x000D_
k(x; u(n))_x000D_
@_x000D_
@u(k) ;_x000D_
for n = 1 we obtain:_x000D_
pr(1)v = _x000D_
@_x000D_
@x_x000D_
+ _x000D_
@_x000D_
@u_x000D_
+ [ x + u0( u x) (u0)2 u]_x000D_
@_x000D_
@u0_x000D_
;_x000D_
then we apply to F(x; u; u0) and equal to zero (this becomes a partial di erential_x000D_
equatio, called conditional symmetry)._x000D_
Then, knowing the vector eld associated with the given di erential equation, we_x000D_
nd their symmetry groups_x000D_
x = (t; x) = etvx =_x000D_
1X_x000D_
k=0_x000D_
tk_x000D_
k!_x000D_
vkx:_x000D_
Finally, we applied the symmetry condition in di erential equations of separable variable,_x000D_
homogeneous equations, exact di erential equations, linear di erential equations,_x000D_
nonlinear di erential equations (Bernoulli and Riccati) applies
Keywords
Ecuaciones diferenciales, Grupos de Lie, Condición de simetría