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% $BNL;RJ*M}3X(B : $B1i=,LdBj(B Q3-2 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B : y9410189 $B;38}(B $B5.9((B
% $B:n@.G/7nF|(B : 95/12/25
% $BEE;R%a!<%k(B : y9410189@edu.cc.uec.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
%

\mondai
{y9410189 $B;38}(B $B5.9((B}
{95/12/25}
{y9410189@edu.cc.uec.ac.jp}
{3-2}
{$B<!$N1i;;;R$N8GM-CM!"8GM-%Y%/%H%k$r5a$a$h!#(B
\begin{enumerate}
\item[(a)]{$ \displaystyle {\cal O} = \frac{d}{dx} + x $ \hspace{10pt} ($B%R%s%H(B: $ \displaystyle \varphi(x) = e^{z(x)} $ $B$H$*$$$F$_$h(B)}
\item[(b)]{$ \displaystyle {\cal O} = \frac{d^2}{dx^2} + \frac{d}{dx} $ \hspace{10pt} ($B%R%s%H(B: $ \displaystyle \varphi(x) = e^{Dx} $ $B$H$*$$$F$_$h(B) }
\end{enumerate}
}
{
(a) $B8GM-CM$r(B $ \lambda $ $B!"(B $B8GM-%Y%/%H%k$r(B $ \varphi(x) $ $B$H$9$k$H!"(B

\begin{equation}
 {\cal O} \varphi(x) = \lambda \varphi(x) , \label{eq:3-2-1}
\end{equation}
${\cal O}$ $B$r(B (\ref{eq:3-2-1}) $B$KBeF~$9$k$H!"(B

%\begin{center}
%  $ \displaystyle  \left( \frac{d}{dx} + x - \lambda \right) \varphi(x) = 0 $
%\end{center}

\begin{equation}
 \frac{d \varphi(x)}{dx} + (x - \lambda) \varphi(x) =0 . \label{eq:3-2-2}
\end{equation}
$B%R%s%H$+$i(B $ \displaystyle \varphi(x) = e^{z(x)} $ $B$H$*$-!"(B
$B$3$l$r(B (\ref{eq:3-2-2}) $B$KBeF~$9$k$H!"(B

\begin{center}
 $ \displaystyle z^{'}(x) e^{z(x)} + (x - \lambda)e^{z(x)} = 0 , $

 $ \displaystyle z^{'}(x) = -x + \lambda , $

 $ \displaystyle z(x) = -\frac{x^2}{2} + \lambda x + C^{'} , $ \hspace{2mm} ($C^{'}$ $B$OG$0UDj?t(B)
\end{center}

$B$7$?$,$C$F(B

\begin{equation}
 \varphi(x) = C e^{-x^2/2 + \lambda x} \label{eq:3-2-3} 
\end{equation}
$B$rF@$k(B($C$ $B$OG$0UDj?t(B)$B!#(B$ \lambda $ $B$OG$0U$N?t$G$"$k$+$i!"8GM-CM!"8GM-4X?t$NAH$,L5?t$K$"$k!#(B \\

(b) (a)$B$HF1MM$K$7$F(B $ {\cal O} $ $B$r(B (\ref{eq:3-2-1}) $B$KBeF~$9$k$H!"(B

\begin{equation}
 \left(\frac{d^2}{dx^2} + \frac{d}{dx} - \lambda \right) \varphi(x) = 0 . \label{eq:3-2-4}
\end{equation}
$B%R%s%H$+$i(B $ \displaystyle \varphi(x) = e^{Dx} $ $B$H$*$-!"$3$l$r(B (\ref{eq:3-2-4}) $B$KBeF~$7$F!"(B

\begin{center}
 $ \displaystyle D^2 + D - \lambda = 0 $

 $ \displaystyle D = \frac{-1 \pm \sqrt{1 + 4 \lambda}}{2} $
\end{center}
$B$7$?$,$C$F!"(B

\begin{equation}
 \varphi = C_{1} e^{(-1 - \sqrt{1 + 4 \lambda})x/2} + C_{2} e^{(-1 + \sqrt{1 + 4 \lambda})x/2} .
\end{equation}
$B$rF@$k(B($C_{1},C_{2}$ $B$OG$0UDj?t(B)$B!#(B
}
