二階微分のラプラス変換

ラプラス変換

$$\begin{array}{rcl}\mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}f\left(t\right)e^{–st}\mathrm{d}t\end{array}$$

二階微分のラプラス変換

$$\begin{array}{rcl}f\left(t\right)& =& \frac{{\mathrm{d}}^{2}g\left(t\right)}{{\mathrm{d}}^{2}t}\\ \mathfrak{L}\left[f\left(t\right)\right]& =& {\int }_0^{\mathrm{\infty }}f\left(t\right)e^{–st}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}\frac{{\mathrm{d}}^{2}g\left(t\right)}{{\mathrm{d}}^{2}t}{e}^{–st}\mathrm{d}t\\ & =& {\left[\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\right]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}(-s)e^{–st}\mathrm{d}t\\ & & \dots {\int }_a^b{f}^{\prime }\left(t\right)g\left(t\right)\mathrm{d}t={\left[f\left(t\right)g\left(t\right)\right]}_a^b-{\int }_a^{b}f\left(t\right)g^{\prime }\left(t\right)\mathrm{d}t(f^{\prime }:f\mathrm{の}\mathrm{微}\mathrm{分},g^{\prime }:g\mathrm{の}\mathrm{部}\mathrm{分})\\ & =& [\frac{\mathrm{d}g\left(\mathrm{\infty }\right)}{\mathrm{d}t}{e}^{–s\mathrm{\infty }}-\left\{{\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}|}_{t=0}\right\}{e}^{–s0}]-(-s){\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & =& [0-g^{\prime }\left(0\right)]+s{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t\\ & & \dots {\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}|}_{t=0}=g^{\prime }\left(0\right)\\ & =& s{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t-g^{\prime }\left(0\right)\\ & =& s\{s\mathfrak{L}[g(t)]-g\left(0\right)\}-g^{\prime }\left(0\right)\\ & & \dots {\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e}^{–st}\mathrm{d}t=\mathfrak{L}\left[\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\right]=s\mathfrak{L}[g(t)]-g\left(0\right)\\ & =& s^2\mathfrak{L}\left[g\left(t\right)\right]-sg\left(0\right)-g^{\prime }\left(0\right)\end{array}$$