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:

   \[
   = \frac{2s(s^2 + 9) \left( -s^2 - 9 + 2s^2 - 18 \right)}{(s^2 + 9)^4}
   = \frac{2s(s^2 + 9) \left( s^2 - 27 \right)}{(s^2 + 9)^4}
   \]

   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.

Discover how to calculate the Laplace transform of t^2 cos(3t) step-by-step. This problem involves using the Laplace transform formula for t^n f(t) and applying the quotient rule for differentiation. Suitable for advanced mathematics enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation