This problem involves using Laplace transforms to solve for $y(t)$. Let's break it down step by step:

1. Laplace Transform and Decomposition:

   • The function $L(y) = \frac{2s}{(s^2 + 4)(4s^2 + 5s + 17)}$ has been decomposed into a product of two separate terms: [ L(y) = \mathcal{L}^{-1}\left[\frac{2s}{s^2 + 4}\right] * \mathcal{L}^{-1}\left[\frac{1}{4s^2 + 5s + 17}\right]. ]

2. Separate Components in the Time Domain:

   • $f(t) = \mathcal{L}^{-1}\left[\frac{2s}{s^2 + 4}\right] = 2\cos(2t)$.
   • $g(t) = \mathcal{L}^{-1}\left[\frac{1}{4s^2 + 5s + 17}\right]$ is derived using inverse Laplace transform techniques. It is given as: $g(t) = \frac{2}{\sqrt{247}} e^{-\frac{5t}{8}} \sin\left(\frac{\sqrt{247}}{8}t\right).$

3. Convolution in the Time Domain:

   • Using the convolution theorem: $y(t) = (f * g)(t) = \int_0^t f(w) g(t-w) \, dw.$
   • Substituting $f(w) = 2\cos(2w)$ and $g(t-w) = \frac{2}{\sqrt{247}} e^{-\frac{5}{8}(t-w)} \sin\left(\frac{\sqrt{247}}{8}(t-w)\right)$, the convolution integral becomes: $y(t) = \int_0^t 2\cos(2w) \cdot \frac{2}{\sqrt{247}} e^{-\frac{5}{8}(t-w)} \sin\left(\frac{\sqrt{247}}{8}(t-w)\right) \, dw.$

4. Simplified Form of the Solution:

   • Factoring constants out of the integral, the solution is expressed as: $y(t) = \frac{4}{\sqrt{247}} \int_0^t \cos(2w) e^{-\frac{5}{8}(t-w)} \sin\left(\frac{\sqrt{247}}{8}(t-w)\right) \, dw.$

This final expression for $y(t)$ demonstrates the application of the convolution theorem.

Let me know if you'd like further steps for solving the integral explicitly or understanding any specific part of this process! Here are five questions for additional exploration:

1. How can the convolution theorem simplify Laplace transform problems in general?
2. What methods are used to compute the inverse Laplace transform for non-trivial terms like $\frac{1}{4s^2 + 5s + 17}$?
3. How does damping influence the $g(t)$ function in this problem?
4. What role does the parameter $\frac{\sqrt{247}}{8}$ play in defining the oscillatory behavior of $g(t)$?
5. How can numerical methods be applied to evaluate the convolution integral for specific values of $t$?

Tip: When solving Laplace problems involving convolution, always focus on simplifying the individual terms in the frequency domain before proceeding to the time domain.