Equacion

Una equacion^[1] (dal latin aequatio^[2]) es una egalitat entre doas expressions contenent una o mantuna variablas, dichas desconegudas. L'utilizacion del tèrme remonta almens al Libèr abbaci de Leonardo de Pisa (1228).

Descripcion

Una equacion es compausat de dos membres d'egalitat ─ de dos costats del signat egal. La solucion d'equacion es la valor de la desconeguda per que los membres equacionaris respècten l'egalitat, qu'es vertadièra o inexistenta.

Principi d'equivaléncia

Dos membres d'una equacion son diches equivalent per l'ensemble de las solucions. Existisson dos principis per resòlvre una equacion e trobar l'ensemble de las solucions de la valor de la variabla, consequéncia de las proprietats d'egalitat:

Primièr principe d'equivaléncia: es pausada una equacion, dont es addicionat o tirat a l'encòp als dos membres equacionaris lo meteis nombre o una meteissa inconeguda, per tal d'obtenir una equacion equivalenta:

Exemple:

$12x+3=10$

$\Leftrightarrow 12x+3-3=10-3$

$\Leftrightarrow 12x=7$

Segond principe d'equivaléncia: es pausada una equacion, dont es multiplicat o dividit a l'encòp als dos membres equacionaris un nombre levat zèro^[3]:

Exemple

$5x+7={\frac {3x}{8}}$

$\Leftrightarrow (5x+7)8=({\frac {3x}{8}})8$

$\Leftrightarrow 40x+56=3$

$8$ es la valor qu'anulla lo denominator ${\frac {3x}{8}}$ .

Notacion

Dins una equacion apareisson, en mai de las inconegudas, dels coeficients coneguts que multiplican las inconegudas elas meteissas e dels tèrmes coneguts que lor son aplicats pel mejan d'una soma algebrica : aqueles elements, se son pas explicits dins lor valor numerica, son generalament indicat per las letras per, $a,b,c$ ... mentre que las darrièras letras de l'alfabet son classicament atribuïdas a las inconegudas $x,y,z$ .

Classificacion

Una primièra classificacion de las equacions pòt aver luòc coma seguís :

las equacions algebricas, que remontan als polinòmis ;

equacions transcendentalas, non reductiblas a de polinòmis ;

equacions a valors absoludas ;

equacions foncionalas, dins las qualas las inconegudas son de foncions.

Equacions algebricas

Las equacions algebricas pòdon èsser divididas en divèrses grops segon lors caracteristicas; se cal remembrar qu'una equacion deu apartenir a almens e una sola de las categorias per cada grop.

Segon lo gra del polinòmi:

equacions del 1èr gra o equacions linearas;

equacions del 2ème gra o equacions quadraticas;

equacions del 3en gra o equacions cubicas;

equacions del 4en gra o equacions quartiques;

equacions del 5en gra o equacions quintiques;

Bibliografia

(en) Renardy, Michael; Rogers. An Introduction to Partial Differential Equations. ISBN 0387004440.
(en) Hale, Jack K.; Verduyn Lunel, Sjoerd M.. Introduction to Functional Differential Equations.

Nòtas e referéncias

↑ Lo Congrès. «Diccionari occitan ─ Dicod'Òc» (en oc, fr).
↑ «Etimologia de «Equation» en francés» (en francés). www.larousse.fr.
↑ Una equacion de tipe ${\frac {x}{0}}$ es non determinada, es impossibla de pausar. Son quocient es pasmens l'infinit.

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