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The image contains two problems where you are asked to find the inverse Laplace transform using the convolution theorem. Let’s break down the problems:

### Problem 94:
\[ \mathcal{L}^{-1} \left( \frac{s^2}{(s^2 + a^2)^2} \right) \]

The given answer is:
\[
\frac{1}{2a} \left[ \sin(at) + at \cos(at) \right]
\]

This is the result after applying the inverse Laplace transform, possibly using the convolution theorem. The form \( (s^2 + a^2) \) suggests a trigonometric function in the time domain.

### Problem 95:
\[ \mathcal{L}^{-1} \left( \frac{s^2 + 2s + 3}{(s^2 + 2s + 5)(s^2 + 2s + 2)} \right) \]

The given answer is:
\[
\frac{e^{-t}}{3} \left( \sin(2t) + \sin(t) \right)
\]

This is a more complicated rational function involving quadratic factors in the denominator, and after applying the convolution theorem, this is the inverse Laplace transform.

Both problems make use of standard techniques from Laplace transforms and convolution. If you would like to see the detailed solution for one or both problems, I can walk you through the steps.

Let me know if you'd like more details or clarification on any of these. Here are 5 related questions you might find helpful:

1. How do we apply the convolution theorem in Laplace transforms?
2. What are the standard Laplace transforms for \( \sin(at) \) and \( \cos(at) \)?
3. What are some common methods for finding inverse Laplace transforms of rational functions?
4. How can we simplify the expression in Problem 95 before applying the convolution theorem?
5. How does the exponential term \( e^{-t} \) appear in inverse Laplace transforms?

**Tip**: The convolution theorem is often helpful when you have products of functions in the Laplace domain. To apply it effectively, ensure the factors can be easily transformed back individually.

Find the inverse of the following using convolution theorem: s^2 / (s^2 + a^2)^2 and (s^2 + 2s + 3) / [(s^2 + 2s + 5)(s^2 + 2s + 2)].

Find the inverse Laplace transforms using the convolution theorem for functions like s^2 / (s^2 + a^2)^2 and (s^2 + 2s + 3) / [(s^2 + 2s + 5)(s^2 + 2s + 2)]. This involves trigonometric functions and the convolution of transformed functions. Suitable for advanced high school or college students.

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