Consider a real-valued function $f(t)$ for a real variable $t$ defined for $0 \leq t < \infty$. The Laplace transform is defined as

$$\mathcal{L}[f(t)] = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, \mathrm{d}t \tag{1}$$

where $s$ is a complex variable whose real part is positive. Under the condition that the improper integral in the right-hand side does not diverge, answer the following questions.

\begin{enumerate}
  \item When the conditions
$$\lim_{t \rightarrow \infty} e^{-st} f(t) = 0 \quad \text{and} \quad \lim_{t \rightarrow \infty} e^{-st} f^{\prime}(t) = 0$$
are satisfied, show the following equation holds:
$$\mathcal{L}\left[f^{\prime\prime}(t)\right] = -f^{\prime}(0) - s f(0) + s^{2} \mathcal{L}[f(t)]$$
Note that $f^{\prime}(t)$ and $f^{\prime\prime}(t)$ are defined as
$$f^{\prime}(t) = \frac{\mathrm{d}f(t)}{\mathrm{d}t} \quad \text{and} \quad f^{\prime\prime}(t) = \frac{\mathrm{d}^{2}f(t)}{\mathrm{d}t^{2}}$$

  \item Calculate the Laplace transform of $g(t) = e^{-at}\cos(\omega t)$ and $h(t) = e^{-at}\sin(\omega t)$ defined for $0 \leq t < \infty$ by showing derivation processes using Equation (1). Note that $a$ and $\omega$ are positive real numbers.

  \item Solve the differential equation
$$f^{\prime\prime}(t) + 6f^{\prime}(t) + 13f(t) = 0$$
where the initial values are $f(0) = 5$ and $f^{\prime}(0) = -11$.
\end{enumerate}