To find the Laplace transform of the function $t^2 \cos(3t)$, we use the formula for the Laplace transform of $t^n f(t)$:

[ \mathcal{L}[t^n f(t)] = (-1)^n \frac{d^n}{ds^n} \left{ \mathcal{L}[f(t)] \right} ]

Here, $f(t) = \cos(3t)$ and $n = 2$. The Laplace transform of $\cos(3t)$ is:

$\mathcal{L}[\cos(3t)] = \frac{s}{s^2 + 9}$

Now, we need to differentiate this result twice with respect to $s$.

Step-by-step Solution:

1. First Derivative: $\frac{d}{ds} \left( \frac{s}{s^2 + 9} \right)$

   Using the quotient rule, $\frac{d}{ds} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}$:

   • $u = s$, $u' = 1$
   • $v = s^2 + 9$, $v' = 2s$

   Plugging these into the formula:

   $\frac{d}{ds} \left( \frac{s}{s^2 + 9} \right) = \frac{(1)(s^2 + 9) - (s)(2s)}{(s^2 + 9)^2} = \frac{s^2 + 9 - 2s^2}{(s^2 + 9)^2} = \frac{-s^2 + 9}{(s^2 + 9)^2}$

2. Second Derivative: $\frac{d^2}{ds^2} \left( \frac{s}{s^2 + 9} \right) = \frac{d}{ds} \left( \frac{-s^2 + 9}{(s^2 + 9)^2} \right)$

   Applying the quotient rule again:

   • $u = -s^2 + 9$, $u' = -2s$
   • $v = (s^2 + 9)^2$, $v' = 2(s^2 + 9)(2s) = 4s(s^2 + 9)$

   Plugging into the quotient rule:

   $\frac{d}{ds} \left( \frac{-s^2 + 9}{(s^2 + 9)^2} \right) = \frac{(-2s)(s^2 + 9)^2 - (-s^2 + 9)(4s(s^2 + 9))}{(s^2 + 9)^4}$

   Simplifying this expression:

   $= \frac{-2s(s^2 + 9)^2 + 4s(s^2 + 9)(s^2 - 9)}{(s^2 + 9)^4}$

   Factor out $2s(s^2 + 9)$:

   $= \frac{2s(s^2 + 9) \left( - (s^2 + 9) + 2(s^2 - 9) \right)}{(s^2 + 9)^4}$

   Simplifying inside the parentheses:

   = \frac{2s(s^2 + 9) \left( s^2 - 27 \right)}{(s^2 + 9)^4}$$ Finally, divide by $$(s^2 + 9)^2$$: $$\frac{2s(s^2 - 27)}{(s^2 + 9)^3}$$

3. Final Result:

The Laplace transform of $t^2 \cos(3t)$ is:

$\mathcal{L}[t^2 \cos(3t)] = \frac{2s(s^2 - 27)}{(s^2 + 9)^3}$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the Laplace transform of $t^3 \sin(2t)$?
2. How does the Laplace transform change for $t^n e^{at} \cos(bt)$?
3. What is the Laplace transform of $t \sin(at)$?
4. How can we apply the Laplace transform to solve higher-order differential equations?
5. What is the inverse Laplace transform of $\frac{s^2 - 9}{(s^2 + 9)^3}$?

Tip: For functions involving $t^n$ multiplied by another function, the differentiation property of the Laplace transform can simplify finding the transform by reducing it to differentiating simpler transforms.