次の各式を計算せよ。
\begin{align}
(1)& (D^2 + D)(x^3 + 2 x) \\
(2)& (D^4 – D^2 + 2) \cos 3x \\
(3)& (D^3 + D^2 – 1) {\rm e}^{2 x} \\
(4)& (D^2 + 2 D + 1) \log x
\end{align}
(1)
\begin{align}
(D^2 + D)(x^3 + 2 x) &= D^2(x^3 + 2 x) + D(x^3 + 2 x) \\
&= 6 x + 2 x^2 + 2
\end{align}
(2)
\begin{align}
(D^4 – D^2 + 2) \cos 3 x &= D^4 \cos 3x – D^2 \cos 3x + 2 \cos 3x \\
&= 81 \cos 3x + 9 \cos 3x + 2 \cos 3x \\
&= 92 \cos 3x
\end{align}
(3)
\begin{align}
(D^3 + D^2 – 1) {\rm e}^{2 x} &= D^3 {\rm e}^{2 x} + D^2 {\rm e}^{2 x} – {\rm e}^{2 x} \\
&= 8 {\rm e}^{2 x} + 4 {\rm e}^{2 x} – {\rm e}^{2 x} \\
&= 11 {\rm e}^{2 x}
\end{align}
(4)
\begin{align}
(D^2 + 2 D + 1)\log x &= D^2 \log x + 2 D \log x + \log x \\
&= – \frac{1}{x^2} + \frac{2}{x} + \log x
\end{align}
