To find the Laplace transform of \( t^2 \cos(3t) \), we can use the Laplace transform properties.

### General Formula:

The Laplace transform of \( t^n f(t) \) is given by:

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

where \( \mathcal{L}\{f(t)\} \) is the Laplace transform of the function \( f(t) \).

### Step 1: Laplace Transform of \( \cos(3t) \)

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

### Step 2: Apply the formula for \( t^2 \cos(3t) \)

Now, since \( n = 2 \) and \( f(t) = \cos(3t) \):

\[
\mathcal{L}\{t^2 \cos(3t)\} = (-1)^2 \frac{d^2}{ds^2} \left[\frac{s}{s^2 + 9}\right]
\]

### Step 3: Differentiate Twice

Let's differentiate \( \frac{s}{s^2 + 9} \) twice with respect to \( s \):

1. **First derivative**:
   \[
   \frac{d}{ds} \left[\frac{s}{s^2 + 9}\right] = \frac{(s^2 + 9) \cdot 1 - s \cdot 2s}{(s^2 + 9)^2} = \frac{9 - s^2}{(s^2 + 9)^2}
   \]

2. **Second derivative**:
   \[
   \frac{d^2}{ds^2} \left[\frac{s}{s^2 + 9}\right] = \frac{d}{ds} \left[\frac{9 - s^2}{(s^2 + 9)^2}\right]
   \]
   Using the quotient rule again:
   \[
   \frac{d}{ds} \left[\frac{9 - s^2}{(s^2 + 9)^2}\right] = \frac{(s^2 + 9)^2 \cdot (-2s) - (9 - s^2) \cdot 2(s^2 + 9) \cdot 2s}{(s^2 + 9)^4}
   \]
   Simplifying this yields:
   \[
   \frac{-2s(s^2 + 9)^2 - 2s(9 - s^2) \cdot 2(s^2 + 9)}{(s^2 + 9)^4}
   \]
   After further simplification:
   \[
   = \frac{2s(3s^2 - 45)}{(s^2 + 9)^3}
   \]

### Final Answer:

\[
\mathcal{L}\{t^2 \cos(3t)\} = \frac{2s(3s^2 - 45)}{(s^2 + 9)^3}
\]

Do you want to go over this in more detail or have any questions?

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Here are 5 related questions:

1. How would you find the Laplace transform of \( t \sin(2t) \)?
2. Can you compute \( \mathcal{L}^{-1} \left\{ \frac{s}{s^2 + 4} \right\} \)?
3. What is the Laplace transform of \( t^3 e^{2t} \)?
4. How can you apply the convolution theorem using Laplace transforms?
5. Can you determine the Laplace transform of a periodic function?

**Tip:** The Laplace transform is particularly useful in solving linear differential equations and analyzing systems.

Explore how to calculate the Laplace transform of t^2 cos(3t) using differentiation and trigonometric functions. This problem involves advanced mathematical concepts in Laplace transforms and is suitable for advanced mathematics students and professionals.

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