. . . . . . . . "In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel. A generalisation to first-order systems of homogeneous linear differential equations is given by Liouville's formula."@en . . . . "In matematica, l'identit\u00E0 di Abel (chiamata anche identit\u00E0 di equazione differenziale di Abel) \u00E8 un'equazione che esprime il Wronskiano di due soluzioni omogenee di un'equazione differenziale lineare del secondo ordine in termini di coefficienti dell'equazione differenziale originale. L'identit\u00E0 prende il nome dal matematico Niels Henrik Abel."@it . "3261224"^^ . "Identidad de Abel"@es . . "Identit\u00E0 di Abel"@it . . . . . "Abelsche Identit\u00E4t"@de . . . . . . . . . . "AbelsDifferentialEquationIdentity"@en . . "Die abelsche Identit\u00E4t ist ein Ausdruck f\u00FCr die Wronski-Determinante zweier linear unabh\u00E4ngiger homogener L\u00F6sungen einer linearen gew\u00F6hnlichen Differentialgleichung zweiter Ordnung. Die Beziehung wurde 1827 von dem norwegischen Mathematiker Niels Henrik Abel (1802\u20131829) hergeleitet."@de . . . . "1092152351"^^ . . . . . . . . . . "Identitat abeliana"@ca . . . . . . . . . . "Die abelsche Identit\u00E4t ist ein Ausdruck f\u00FCr die Wronski-Determinante zweier linear unabh\u00E4ngiger homogener L\u00F6sungen einer linearen gew\u00F6hnlichen Differentialgleichung zweiter Ordnung. Die Beziehung wurde 1827 von dem norwegischen Mathematiker Niels Henrik Abel (1802\u20131829) hergeleitet."@de . "In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation.The relation can be generalised to nth-order linear ordinary differential equations. The identity is named after the Norwegian mathematician Niels Henrik Abel. Since Abel's identity relates the different linearly independent solutions of the differential equation, it can be used to find one solution from the other. It provides useful identities relating the solutions, and is also useful as a part of other techniques such as the method of variation of parameters. It is especially useful for equations such as Bessel's equation where the solutions do not have a simple analytical form, because in such cases the Wronskian is difficult to compute directly. A generalisation to first-order systems of homogeneous linear differential equations is given by Liouville's formula."@en . . . . . . "La identidad de Abel es una expresi\u00F3n matem\u00E1tica utilizada para calcular el Wronskiano de dos funciones de una ecuaci\u00F3n diferencial. Sea una ecuaci\u00F3n diferencial donde es una constante que se hallar\u00E1 con las condiciones iniciales dadas. Ejemplo: Con la condici\u00F3n dada se halla \n* Datos: Q318714"@es . . . . "10970"^^ . . . "Abel's identity"@en . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0623\u0628\u064A\u0644 (\u0648\u062A\u0633\u0645\u0649 \u0623\u064A\u0636\u0627 \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0623\u0628\u064A\u0644 \u062D\u0648\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629) \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0639\u0628\u0631 \u0639\u0646 \u0644\u062D\u0644\u064A\u0646 \u0645\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0639\u0627\u062F\u064A\u0629 \u062E\u0637\u064A\u0629 \u0645\u062A\u062C\u0627\u0646\u0633\u0629 \u0645\u0646 \u0627\u0644\u062F\u0631\u062C\u0629 \u0627\u0644\u062B\u0627\u0646\u064A\u0629 \u0628\u062F\u0644\u0627\u0644\u0629 \u0645\u0639\u0627\u0645\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u0623\u0635\u0644\u064A\u0629. \u064A\u0645\u0643\u0646 \u062A\u0639\u0645\u064A\u0645 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u062E\u0637\u064A\u0629 \u0627\u0644\u0639\u0627\u062F\u064A\u0629 \u0645\u0646 \u0627\u0644\u0631\u062A\u0628\u0629 n. \u0633\u0645\u064A\u062A \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0646\u0633\u0628\u0629\u064B \u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0627\u0644\u0646\u0631\u0648\u064A\u062C\u064A \u0646\u064A\u0644\u0632 \u0647\u0646\u0631\u064A\u0643 \u0623\u0628\u064A\u0644."@ar . "\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0623\u0628\u064A\u0644"@ar . . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0623\u0628\u064A\u0644 (\u0648\u062A\u0633\u0645\u0649 \u0623\u064A\u0636\u0627 \u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0623\u0628\u064A\u0644 \u062D\u0648\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629) \u0647\u064A \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0639\u0628\u0631 \u0639\u0646 \u0644\u062D\u0644\u064A\u0646 \u0645\u0646 \u0645\u0639\u0627\u062F\u0644\u0629 \u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0639\u0627\u062F\u064A\u0629 \u062E\u0637\u064A\u0629 \u0645\u062A\u062C\u0627\u0646\u0633\u0629 \u0645\u0646 \u0627\u0644\u062F\u0631\u062C\u0629 \u0627\u0644\u062B\u0627\u0646\u064A\u0629 \u0628\u062F\u0644\u0627\u0644\u0629 \u0645\u0639\u0627\u0645\u0644 \u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0629 \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u0623\u0635\u0644\u064A\u0629. \u064A\u0645\u0643\u0646 \u062A\u0639\u0645\u064A\u0645 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u0627\u0644\u0645\u0639\u0627\u062F\u0644\u0627\u062A \u0627\u0644\u062A\u0641\u0627\u0636\u0644\u064A\u0629 \u0627\u0644\u062E\u0637\u064A\u0629 \u0627\u0644\u0639\u0627\u062F\u064A\u0629 \u0645\u0646 \u0627\u0644\u0631\u062A\u0628\u0629 n. \u0633\u0645\u064A\u062A \u0627\u0644\u0645\u062A\u0637\u0627\u0628\u0642\u0629 \u0646\u0633\u0628\u0629\u064B \u0644\u0639\u0627\u0644\u0645 \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A \u0627\u0644\u0646\u0631\u0648\u064A\u062C\u064A \u0646\u064A\u0644\u0632 \u0647\u0646\u0631\u064A\u0643 \u0623\u0628\u064A\u0644."@ar . "En matem\u00E0tiques, la identitat abeliana \u00E9s una equaci\u00F3 que expressa el Wronski\u00E0 de dues solucions homog\u00E8nies d'una equaci\u00F3 diferencial ordin\u00E0ria lineal de segon ordre en termes dels coeficients de l'equaci\u00F3 diferencial original. La identitat deu el seu nom al matem\u00E0tic Niels Henrik Abel. La identitat abeliana, en relacionar les solucions linealment independents d'una equaci\u00F3 diferencial, es pot fer servir per trobar una soluci\u00F3 de l'altra, proporciona identitats \u00FAtils relacionades amb les solucions, i tamb\u00E9 \u00E9s \u00FAtil com a part d'altres t\u00E8cniques com el m\u00E8tode de variaci\u00F3 dels par\u00E0metres. \u00C9s especialment \u00FAtil en equacions com l'equaci\u00F3 de Bessel, en la qual les solucions no tenen una forma anal\u00EDtica simple, ja que en aquests casos el Wronski\u00E0 \u00E9s dif\u00EDcil de calcular directament."@ca . . . "La identidad de Abel es una expresi\u00F3n matem\u00E1tica utilizada para calcular el Wronskiano de dos funciones de una ecuaci\u00F3n diferencial. Sea una ecuaci\u00F3n diferencial donde es una constante que se hallar\u00E1 con las condiciones iniciales dadas. Ejemplo: Con la condici\u00F3n dada se halla \n* Datos: Q318714"@es . . "In matematica, l'identit\u00E0 di Abel (chiamata anche identit\u00E0 di equazione differenziale di Abel) \u00E8 un'equazione che esprime il Wronskiano di due soluzioni omogenee di un'equazione differenziale lineare del secondo ordine in termini di coefficienti dell'equazione differenziale originale. L'identit\u00E0 prende il nome dal matematico Niels Henrik Abel. L'identit\u00E0 di Abel, siccome si riferisce a diverse soluzioni linearmente indipendenti dell'equazione differenziale, pu\u00F2 essere usata per trovare una soluzione partendo dall'altra. \u00C8 molto utile per equazioni come le equazioni di Bessel, dove le soluzioni non hanno una forma analitica, poich\u00E9 in quei casi il Wronskiano \u00E8 difficile da calcolare direttamente."@it . "Abel's Differential Equation Identity"@en . . . . "En matem\u00E0tiques, la identitat abeliana \u00E9s una equaci\u00F3 que expressa el Wronski\u00E0 de dues solucions homog\u00E8nies d'una equaci\u00F3 diferencial ordin\u00E0ria lineal de segon ordre en termes dels coeficients de l'equaci\u00F3 diferencial original. La identitat deu el seu nom al matem\u00E0tic Niels Henrik Abel."@ca . .