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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 annihilator method ： ウィキペディア英語版 annihilator method In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations (ODE's). It is similar to the method of undetermined coefficients, but instead of guessing the particular solution in the method of undetermined coefficients, the particular solution is determined systematically in this technique. The phrase ''undetermined coefficients'' can also be used to refer to the step in the annihilator method in which the coefficients are calculated. The annihilator method is used as follows. Given the ODE $P(D)y=f(x)$, find another differential operator $A(D)$ such that $A(D)f(x)\; =\; 0$. This operator is called the annihilator, thus giving the method its name. Applying $A(D)$ to both sides of the ODE gives a homogeneous ODE $\backslash big(A(D)P(D)\backslash big)y\; =\; 0$ for which we find a solution basis $\backslash$ as before. Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combination to satisfy the ODE. This method is not as general as variation of parameters in the sense that an annihilator does not always exist. ==Example== Given $y\text{'}\text{'}-4y\text{'}+5y=\backslash sin(kx)$, $P(D)=D^2-4D+5$. The simplest annihilator of $\backslash sin(kx)$ is $A(D)=D^2+k^2$. The zeros of $A(z)P(z)$ are $\backslash$, so the solution basis of $A(D)P(D)$ is $\backslash =\backslash ,e^,e^\backslash \}.$ Setting $y=c\_1y\_1+c\_2y\_2+c\_3y\_3+c\_4y\_4$ we find : $$ \begin \sin(kx) & = P(D)y \\() & = P(D)(c_1y_1+c_2y_2+c_3y_3+c_4y_4) \\() & =c_1P(D)y_1+c_2P(D)y_2+c_3P(D)y_3+c_4P(D)y_4 \\() & =0+0+c_3(-k^2-4ik+5)y_3+c_4(-k^2+4ik+5)y_4 \\() & =c_3(-k^2-4ik+5)(\cos(kx)+i\sin(kx)) +c_4(-k^2+4ik+5)(\cos(kx)-i\sin(kx)) \end giving the system :$i=(k^2+4ik-5)c\_3+(-k^2+4ik+5)c\_4$ :$0=(k^2+4ik-5)c\_3+(k^2-4ik-5)c\_4$ which has solutions :$c\_3=\backslash frac\; i$, $c\_4=\backslash frac\; i$ giving the solution set : $$ \begin y & = c_1y_1+c_2y_2+\frac iy_3+\frac iy_4 \\() & =c_1y_1+c_2y_2+\frac \\() & =c_1y_1+c_2y_2+\frac. \end This solution can be broken down into the homogeneous and nonhomogeneous parts. In particular, $y\_p\; =\; \backslash frac$ is a particular integral for the nonhomogeneous differential equation, and $y\_c\; =\; c\_1y\_1\; +\; c\_2y\_2$ is a complementary solution to the corresponding homogeneous equation. The values of $c\_1$ and $c\_2$ are determined usually through a set of initial conditions. Since this is a second-order equation, two such conditions are necessary to determine these values. The fundamental solutions $y\_1\; =\; e^$ and $y\_2\; =\; e^$ can be further rewritten using Euler's formula: : $e^\; =\; e^\; e^\; =\; e^\; (\backslash cos\; x\; +\; i\; \backslash sin\; x)$ : $e^\; =\; e^\; e^\; =\; e^\; (\backslash cos\; x\; -\; i\; \backslash sin\; x)$ Then $c\_1\; y\_1\; +\; c\_2\; y\_2\; =\; c\_1e^\; (\backslash cos\; x\; +\; i\; \backslash sin\; x)\; +\; c\_2\; e^\; (\backslash cos\; x\; -\; i\; \backslash sin\; x)\; =\; (c\_1\; +\; c\_2)\; e^\; \backslash cos\; x\; +\; i(c\_1\; -\; c\_2)\; e^\; \backslash sin\; x$, and a suitable reassignment of the constants gives a simpler and more understandable form of the complementary solution, $y\_c\; =\; e^\; (c\_1\; \backslash cos\; x\; +\; c\_2\; \backslash sin\; x)$. 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「annihilator method」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.