$$\text { Let } \mathrm { f } ( \mathrm { t } ) = 2 \cos ( 5 \mathrm { t } ) \mathrm { u } ( \mathrm { t } ) \text {. Find the one-sided Laplace transform of } \mathrm { f } ( \mathrm { t } ) , g ( t ) = \frac { d f ( t ) } { d t } \text {, and }$$ $$h ( t ) = \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } }$$

Question

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The Answer is

\begin{array} { l } \mathrm { f } ( 0 ) = 2 \\mathrm {~g} ( \mathrm { t } ) = - 10 \sin ( 5 \mathrm { t } ) , \mathrm { g } ( 0 ) = 0 \\mathrm {~h} ( \mathrm { t } ) = - 50 \cos ( 5 \mathrm { t } ) , \mathrm { h } ( 0 ) = - 50 \F ( s ) = \frac { 2 s } { s ^ { 2 } + 5 ^ { 2 } } \G ( s ) = \frac { - 50 } { s ^ { 2 } + 5 ^ { 2 } } \H ( s ) = \frac { - 50 s } { s ^ { 2 } + 5 ^ { 2 } }\end{array}

h(t)=d2f(t)dt2h ( t ) = \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } }h(t)=dt2d2f(t)​

$h ( t ) = \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } }$