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The problem is asking to find the **Laplace transform** of the **convolution** of two functions, \( f(t) = \cos(2t) \) and \( g(t) = e^{-3t} \).

### Step 1: Convolution of the Two Functions
The convolution of two functions \( f(t) \) and \( g(t) \) is defined as:
\[
(f * g)(t) = \int_0^t f(\tau) g(t - \tau) \, d\tau
\]

However, we don't need to compute this directly since we are asked to find the **Laplace transform** of the convolution.

### Step 2: Laplace Transform of Convolution
One of the properties of the Laplace transform is:
\[
\mathcal{L}\{f * g\}(s) = \mathcal{L}\{f(t)\}(s) \cdot \mathcal{L}\{g(t)\}(s)
\]
This means we can take the Laplace transform of each function individually, then multiply the results.

### Step 3: Laplace Transform of \( f(t) = \cos(2t) \)
Using the Laplace transform table for \( \cos(at) \):
\[
\mathcal{L}\{\cos(2t)\}(s) = \frac{s}{s^2 + 4}
\]

### Step 4: Laplace Transform of \( g(t) = e^{-3t} \)
Using the Laplace transform table for \( e^{-at} \):
\[
\mathcal{L}\{e^{-3t}\}(s) = \frac{1}{s + 3}
\]

### Step 5: Multiply the Laplace Transforms
Now, we multiply the two Laplace transforms:
\[
\mathcal{L}\{f * g\}(s) = \frac{s}{s^2 + 4} \cdot \frac{1}{s + 3}
\]
This gives the final result:
\[
\mathcal{L}\{f * g\}(s) = \frac{s}{(s^2 + 4)(s + 3)}
\]

Thus, the **Laplace transform of the convolution** of the two functions is:
\[
\boxed{\frac{s}{(s^2 + 4)(s + 3)}}
\]

Would you like more details or further clarification on any part of this process?

Here are some related questions for further practice:
1. What is the Laplace transform of \( \sin(2t) \)?
2. How do you compute the convolution \( (f * g)(t) \) directly?
3. What is the inverse Laplace transform of \( \frac{s}{(s^2 + 4)(s + 1)} \)?
4. What is the Laplace transform of \( t \cos(2t) \)?
5. How does the convolution property simplify solving differential equations?

**Tip:** The convolution theorem is a powerful tool for simplifying the analysis of linear systems using Laplace transforms.

Find the Laplace transform of the convolution of the two functions f(t) = cos(2t) and g(t) = e^{-3t}.

Discover how to find the Laplace transform of the convolution of the functions f(t) = cos(2t) and g(t) = e^{-3t}. This solution involves key mathematical concepts such as Laplace transform and convolution, providing a step-by-step explanation suitable for college-level students.

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