If $U ( x , s ) = \mathcal { C } \{ u ( x , t ) \} , \text { then } \left\{ \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } \right\}$ is
A) $U _ { s } ( x , s )$
B) $s U ( x , s )$
C) $s ^ { 2 } U ( x , s ) - u ( x , 0 )$
D) $s ^ { 2 } U ( x , s ) - u ( x , 0 ) - s u _ { t } ( x , 0 )$
E) $s ^ { 2 } U ( x , s ) - s u ( x , 0 ) - u _ { t } ( x , 0 )$

Question

Quizplus · Accepted Answer

The Laplace transform of the second derivative of a function with respect to time, $\frac{\partial^2 u}{\partial t^2}$, is given by $s^2 U(x, s) - s u(x, 0) - u_t(x, 0)$, where $U(x, s)$ is the Laplace transform of $u(x, t)$, $u(x, 0)$ is the initial condition of $u$, and $u_t(x, 0)$ is the initial condition of the first derivative of $u$ with respect to time.

If U(x,s)=C{u(x,t)}, then {∂2u∂t2}U ( x , s ) = \mathcal { C } \{ u ( x , t ) \} , \text { then } \left\{ \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } \right\}U(x,s)=C{u(x,t)}, then {∂t2∂2u​}

If $U ( x , s ) = \mathcal { C } \{ u ( x , t ) \} , \text { then } \left\{ \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } \right\}$