The Laplace transform of $t ^ { 2 } \cos ( 4 t )$ is
A) $\left( 2 / s ^ { 3 } \right) s / \left( s ^ { 2 } + 16 \right)$
B) $\left( 2 / s ^ { 2 } \right) s / \left( s ^ { 2 } + 16 \right)$
C) $8 / \left( s ^ { 3 } \right) \left( s ^ { 2 } + 16 \right)$
D) $\left( 2 s ^ { 3 } - 32 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$
E) $\left( 2 s ^ { 3 } - 96 s \right) / \left( s ^ { 2 } + 16 \right) ^ { 3 }$

Question

Quizplus · Accepted Answer

The Laplace transform of $t^n \cos(at)$ is given by $\frac{n! s^{n-1}}{(s^2 + a^2)^{n+1}}$. For $t^2 \cos(4t)$, $n=2$ and $a=4$, resulting in $\frac{2! (s^2 - 16)}{(s^2 + 16)^3}$, which simplifies to the given option E.

The Laplace Transform Of t2cos⁡(4t)t ^ { 2 } \cos ( 4 t )t2cos(4t)

The Laplace Transform Of $t ^ { 2 } \cos ( 4 t )$