二階微分のラプラス変換

ラプラス変換

$$\begin{eqnarray} \mathfrak{L}\left[ {f\left( t \right)} \right] &=&\int_0^\infty {f\left( t \right){e^{–st}}}\mathrm{d}t \end{eqnarray}$$

二階微分のラプラス変換

$$\begin{eqnarray} f\left(t\right)&=&\frac{\mathrm{d^2}g\left(t\right)}{\mathrm{d^2}t} \\\mathfrak{L}\left[ {f\left( t \right)} \right] &=&\int_0^\infty {f\left( t \right){e^{–st}}}\mathrm{d}t \\&=&\int_0^\infty {\frac{\mathrm{d^2}g\left(t\right)}{\mathrm{d^2}t}{e^{–st}}}\mathrm{d}t \\&=&\left[\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e^{–st}}\right]_0^\infty-\int_0^\infty {\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{\left(-s\right)e^{–st}}}\mathrm{d}t \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post_7.html}{\int_a^b{f'\left( t \right) g\left( t \right) }\mathrm{d}t=\left[f\left( t \right) g\left( t \right)\right]_a^b-\int_a^b{f\left( t \right) g'\left( t \right) }\mathrm{d}t\;\;(f':fの微分,\;g':gの部分)} \\&=&\left[\frac{\mathrm{d}g\left(\infty\right)}{\mathrm{d}t}{e^{–s\infty}} -\left\{\left.\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\right|_{t=0}\right\}{e^{–s0}}\right] -\left(-s\right)\int_0^\infty{\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e^{–st}}}\mathrm{d}t \\&=&\left[0-g^{\prime}\left(0\right)\right] +s\int_0^\infty{\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e^{–st}}}\mathrm{d}t \\&&\;\ldots\;\left.\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\right|_{t=0}=g^{\prime}\left(0\right) \\&=&s\int_0^\infty{\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e^{–st}}}\mathrm{d}t -g^{\prime}\left(0\right) \\&=&s\left\{s\mathfrak{L}\left[g(t)\right]- g\left(0\right)\right\}-g^{\prime}\left(0\right) \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/fracmathrmdfmathrmdt.html}{\int_0^\infty{\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}{e^{–st}}}\mathrm{d}t =\mathfrak{L}\left[\frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}\right] =s\mathfrak{L}\left[g(t)\right]- g\left(0\right)} \\&=&s^2\mathfrak{L}\left[ {g\left( t \right)} \right] - sg\left(0\right) -g^{\prime}\left(0\right) \end{eqnarray}$$