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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 Binomial differential equation ： ウィキペディア英語版 Binomial differential equation The binomial differential equation is the ordinary differential equation : $\backslash left(\; y\text{'}\; \backslash right)^m\; =\; f(x,y),\backslash ,$ when $m$ is a natural number (i.e., a positive integer), and $f(x,y)$ is a polynomial in two variables (i.e., a bivariate polynomial). == The Solution == Let $P\backslash left(\; \backslash right)\; =\; \backslash left(\; \backslash right)^k\; =\; \backslash sum\backslash limits\_^k\; c\}$ k \\ j \\ \end} \right)x^j y^ } be a polynomial in two variables of order $k$; where $k$ is a positive integer. The binomial differential equation becomes $\backslash left(\; \backslash right)^m\; =\; \backslash left(\; \backslash right)^k;$ using the substitution $v\; =\; x\; +\; y$, we get that $v\text{'}\; =\; 1\; +\; y\text{'}$, therefore $\backslash left(\; \backslash right)^m\; =\; v^k$ or we can write $v\text{'}\; =\; 1\; +\; v^\}$, which is a separable ordinary differential equation, hence $$ \frac = 1 + v^} \Rightarrow \frac = dx \Rightarrow \int = x + C. Special cases: - If $m=k$, we have the differential equation $v\text{'}\; -\; 1\; =\; v$ and the solution is $y\backslash left(\; x\; \backslash right)\; =\; Ce^x\; -\; x\; -\; 1$, where $C$ is a constant. - If $m|k$, i.e., $m$ divides $k$ so that there is a positive integer $n$ such that $k=n\; m$, then the solution has the form $\backslash int\; =\; x\; +\; C$. From the tables book of Gradshteyn and Ryzhik we found that $$ \int = \left\ - \frac\sum\limits_^ - 1} \right)} + \frac\sum\limits_^ - 1} \right)} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,n:\,\ln \left( \right) - \frac\sum\limits_^ \\ \end \right. and $$ P_i = \frac\ln \left( \right) + 1} \right), $Q\_i\; =\; \backslash arctan\; \backslash left(\; \backslash right).$ 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「Binomial differential equation」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.