Question

Prove that the laplace transform of t^2 is 2/s^3

Answer 1

well u need to evaluate this integral:\[\int_{0}^{\infty} t^2 e^{-st} dt\]it will be a integration by parts i guess :)

Answer 2

\[L(t)=\int\limits_{0}^{\infty}e ^{-st}tdt\]
\[=(t\frac{e ^{-st}}{-s})_{0}^{\infty}-\int\limits_{0}^{\infty}\frac{e ^{-st}}{-s}dt\]
\[=\frac{1}{s ^{2}}\]
since for s > 0
\[\lim_{t \rightarrow \infty}(te ^{-st})=0\]
Now we have
\[\frac{1}{s ^{2}}=\int\limits_{0}^{\infty}e ^{-st}tdt\]
Differentiating both sides with respect to s
\[-\frac{2}{s ^{3}}=\frac{d}{ds}\int\limits_{0}^{\infty}e ^{-st}tdt\]
\[=\int\limits_{0}^{\infty}\frac{\partial}{\partial s}(e ^{-st})tdt\]
\[=\int\limits_{0}^{\infty}-te ^{-st}tdt\]
so that
\[\frac{2}{s ^{3}}=\int\limits_{0}^{\infty}e ^{-st}t ^{2}dt\]
which gives the Laplace transform of t^2