The Laplace transform of $t ^ { 2 } \sin ( 3 t )$ is
A) $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$
B) $\left( 18 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$
C) $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 2 }$
D) $\left( 30 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$
E) $\left( 12 s ^ { 2 } - 54 \right) / \left( s ^ { 2 } + 9 \right) ^ { 3 }$

Question

Quizplus · Accepted Answer

The Laplace transform of $t^n e^{at} \sin(bt)$ is given by the formula $\frac{n!b}{(s-a)^2 + b^2}^{(n+1)}$. For $t^2 \sin(3t)$, this translates to $\frac{2! \cdot 3}{(s-0)^2 + 3^2}^{3}$, simplifying to $\frac{6}{(s^2 + 9)^3}$. However, the correct form, considering the differentiation property of the Laplace transform, which involves increasing the power of $s$ in the numerator for each differentiation, leads to the form $\frac{18s^2 - 54}{(s^2 + 9)^3}$, matching option A.

The Laplace Transform Of t2sin⁡(3t)t ^ { 2 } \sin ( 3 t )t2sin(3t)

The Laplace Transform Of $t ^ { 2 } \sin ( 3 t )$