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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Abel's identity ： ウィキペディア英語版 :''"Abel's formula" redirects here. For the formula on difference operators, see Summation by parts.''In mathematics, Abel's identity (also called 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 ''n''th-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.==Statement of Abel's identity==Consider a homogeneous linear second-order ordinary differential equation: y'' + p(x)y' + q(x)\,y = 0on an interval ''I'' of the real line with real- or complex-valued continuous functions ''p'' and ''q''. Abel's identity states that the Wronskian ''W''(''y''1,''y''2) of two real- or complex-valued solutions ''y''1 and ''y''2 of this differential equation, that is the function defined by the determinant: W(y_1,y_2)(x)=\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end=y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I,satisfies the relation: W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \exp\biggl(-\int_^x p(\xi) \,\textrm\xi\biggr),\qquad x\in I,for every point ''x''0 in ''I''. :''"Abel's formula" redirects here. For the formula on difference operators, see Summation by parts.'' In mathematics, Abel's identity (also called 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 ''n''th-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. ==Statement of Abel's identity== Consider a homogeneous linear second-order ordinary differential equation : $y\text{'}\text{'}\; +\; p(x)y\text{'}\; +\; q(x)\backslash ,y\; =\; 0$ on an interval ''I'' of the real line with real- or complex-valued continuous functions ''p'' and ''q''. Abel's identity states that the Wronskian ''W''(''y''1,''y''2) of two real- or complex-valued solutions ''y''1 and ''y''2 of this differential equation, that is the function defined by the determinant : $W(y\_1,y\_2)(x)$ =\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end =y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I, satisfies the relation : $W(y\_1,y\_2)(x)=W(y\_1,y\_2)(x\_0)\; \backslash exp\backslash biggl(-\backslash int\_^x\; p(\backslash xi)\; \backslash ,\backslash textrm\backslash xi\backslash biggr),\backslash qquad\; x\backslash in\; I,$ for every point ''x''0 in ''I''. 抄文引用元・出典: フリー百科事典『 Abel's identity (also called 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 ''n''th-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.==Statement of Abel's identity==Consider a homogeneous linear second-order ordinary differential equation: y'' + p(x)y' + q(x)\,y = 0on an interval ''I'' of the real line with real- or complex-valued continuous functions ''p'' and ''q''. Abel's identity states that the Wronskian ''W''(''y''1,''y''2) of two real- or complex-valued solutions ''y''1 and ''y''2 of this differential equation, that is the function defined by the determinant: W(y_1,y_2)(x)=\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end=y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I,satisfies the relation: W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \exp\biggl(-\int_^x p(\xi) \,\textrm\xi\biggr),\qquad x\in I,for every point ''x''0 in ''I''.">ウィキペディア（Wikipedia）』 ■Abel's identity (also called 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 ''n''th-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.==Statement of Abel's identity==Consider a homogeneous linear second-order ordinary differential equation: y'' + p(x)y' + q(x)\,y = 0on an interval ''I'' of the real line with real- or complex-valued continuous functions ''p'' and ''q''. Abel's identity states that the Wronskian ''W''(''y''1,''y''2) of two real- or complex-valued solutions ''y''1 and ''y''2 of this differential equation, that is the function defined by the determinant: W(y_1,y_2)(x)=\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end=y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I,satisfies the relation: W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \exp\biggl(-\int_^x p(\xi) \,\textrm\xi\biggr),\qquad x\in I,for every point ''x''0 in ''I''.">ウィキペディアで「:''"Abel's formula" redirects here. For the formula on difference operators, see Summation by parts.''In mathematics, Abel's identity (also called 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 ''n''th-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.==Statement of Abel's identity==Consider a homogeneous linear second-order ordinary differential equation: y'' + p(x)y' + q(x)\,y = 0on an interval ''I'' of the real line with real- or complex-valued continuous functions ''p'' and ''q''. Abel's identity states that the Wronskian ''W''(''y''1,''y''2) of two real- or complex-valued solutions ''y''1 and ''y''2 of this differential equation, that is the function defined by the determinant: W(y_1,y_2)(x)=\beginy_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end=y_1(x)\,y'_2(x) - y'_1(x)\,y_2(x),\qquad x\in I,satisfies the relation: W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \exp\biggl(-\int_^x p(\xi) \,\textrm\xi\biggr),\qquad x\in I,for every point ''x''0 in ''I''.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.