\section*{Problem 5}
The Laplace transform $F ( s ) = L [ f ( t ) ]$ of a function $f ( t )$, where $t \geq 0$, is defined as

$$F ( s ) = \int _ { 0 } ^ { \infty } f ( t ) e ^ { - s t } d t$$

Here, $s$ is a complex number, and $e$ is the base of the natural logarithm. Answer the following questions. Show the derivation process with your answer.\\
I. Prove the following relations:

\begin{enumerate}
  \item $L \left[ t ^ { n } \right] = \frac { n ! } { s ^ { n + 1 } }$, where $n$ is a natural number.
  \item $L \left[ \frac { d f ( t ) } { d t } \right] = s F ( s ) - f ( 0 )$, where $f ( t )$ is a differentiable function.
  \item $L \left[ e ^ { a t } f ( t ) \right] = F ( s - a )$, where $a$ is a real number.
\end{enumerate}

II. Solve the following differential equation using a Laplace transformation for $t \geq 0$:

$$t \frac { d ^ { 2 } f ( t ) } { d t ^ { 2 } } + ( 1 + 3 t ) \frac { d f ( t ) } { d t } + 3 f ( t ) = 0 , \quad f ( 0 ) = 1 , \left. \quad \frac { d f } { d t } \right| _ { t = 0 } = - 3$$

You can use the relation $L [ t f ( t ) ] = - \frac { d } { d s } F ( s )$, if necessary.\\
III. The point $\mathrm { P } ( x ( t ) , y ( t ) )$, which satisfies the following simultaneous differential equations, passes through the point $( a , b )$ when $t = 0$. $a$ and $b$ are real numbers.

$$\left\{ \begin{array} { l } 
\frac { d x ( t ) } { d t } = - x ( t ) \\
\frac { d y ( t ) } { d t } = x ( t ) - 2 y ( t )
\end{array} \right.$$

\begin{enumerate}
  \item Solve the simultaneous differential equations using a Laplace transformation for $t \geq 0$.
  \item Express the relation between $x$ and $y$ by eliminating $t$ from the solution of III. 1.
  \item For both $( a , b ) = ( 1,1 )$ and $( - 1,1 )$, draw the trajectories of point P when $t$ varies continuously from 0 to infinity.
\end{enumerate}