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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 Characteristic equation (calculus) ： ウィキペディア英語版 Characteristic equation (calculus) In mathematics, the characteristic equation (or auxiliary equation〔) is an algebraic equation of degree $n\; \backslash ,$ on which depends the solutions of a given $n\backslash ,$th-order differential equation. The characteristic equation can only be formed when the differential equation is linear, homogeneous, and has constant coefficients. Such a differential equation, with $y\; \backslash ,$ as the dependent variable and $a\_,\; a\_,\; \backslash ldots\; ,\; a\_,\; a\_$ as constants, :$a\_y^\; +\; a\_y^\; +\; \backslash cdots\; +\; a\_y\text{'}\; +\; a\_y\; =\; 0$ will have a characteristic equation of the form :$a\_r^\; +\; a\_r^\; +\; \backslash cdots\; +\; a\_r\; +\; a\_\; =\; 0$ where $r^,\; r^,\; \backslash ldots\; ,r$ are the roots from which the general solution can be formed.〔 This method of integrating linear ordinary differential equations with constant coefficients was discovered by Leonhard Euler, who found that the solutions depended on an algebraic 'characteristic' equation.〔 The qualities of the Euler's characteristic equation were later considered in greater detail by French mathematicians Augustin-Louis Cauchy and Gaspard Monge.〔〔 == Derivation == Starting with a linear homogeneous differential equation with constant coefficients $a\_,\; a\_,\; \backslash ldots\; ,\; a\_,\; a\_$, :$a\_y^\; +\; a\_y^\; +\; \backslash cdots\; +\; a\_y^\; +\; a\_y\; =\; 0$ it can be seen that if $y(x)\; =\; e^\; \backslash ,$, each term would be a constant multiple of $e^\; \backslash ,$. This results from the fact that the derivative of the exponential function $e^\; \backslash ,$ is a multiple of itself. Therefore, $y\text{'}\; =\; re^\; \backslash ,$, $y\text{'}\text{'}\; =\; r^e^\; \backslash ,$, and $y^\; =\; r^e^\; \backslash ,$ are all multiples. This suggests that certain values of $r\; \backslash ,$ will allow multiples of $e^\; \backslash ,$ to sum to zero, thus solving the homogeneous differential equation.〔 In order to solve for $r\; \backslash ,$, one can substitute $y\; =\; e^\; \backslash ,$ and its derivatives into the differential equation to get :$a\_r^e^\; +\; a\_r^e^\; +\; \backslash cdots\; +\; a\_re^\; +\; a\_e^\; =\; 0$ Since $e^\; \backslash ,$ can never equate to zero, it can be divided out, giving the characteristic equation :$a\_r^\; +\; a\_r^\; +\; \backslash cdots\; +\; a\_r\; +\; a\_\; =\; 0$ By solving for the roots, $r\; \backslash ,$, in this characteristic equation, one can find the general solution to the differential equation.〔〔 For example, if $r\; \backslash ,$ is found to equal to 3, then the general solution will be $y(x)\; =\; ce^\; \backslash ,$, where $c\; \backslash ,$ is an arbitrary constant. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Characteristic equation (calculus)」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.