\section*{Problem 1}
I. Find the general solution $y ( x )$ of the following differential equation:

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 - y ) ,$$

where $0 < y < 1$.

II. Find the value of the following definite integral, $I$ :

$$I = \int _ { - 1 } ^ { 1 } \frac { \arccos \left( \frac { x } { 2 } \right) } { \cos ^ { 2 } \left( \frac { \pi } { 3 } x \right) } \mathrm { d } x$$

where $0 \leq \arccos \left( \frac { x } { 2 } \right) \leq \pi$.

III. For any positive variable $x$, we define $f ( x )$ and $g ( x )$ respectively as

$$f ( x ) = \sum _ { m = 0 } ^ { \infty } \frac { 1 } { ( 2 m ) ! } x ^ { 2 m }$$

and

$$g ( x ) = \frac { \mathrm { d } } { \mathrm {~d} x } f ( x )$$

For any non-negative integer $n , I _ { n } ( x )$ is defined as

$$I _ { n } ( x ) = \int _ { 0 } ^ { x } \left\{ \frac { g ( X ) } { f ( X ) } \right\} ^ { n } \mathrm {~d} X$$

Here, you may use

$$\exp ( x ) = \sum _ { m = 0 } ^ { \infty } \frac { 1 } { m ! } x ^ { m }$$

\begin{enumerate}
  \item Calculate $f ( x ) ^ { 2 } - g ( x ) ^ { 2 }$.
  \item Express $I _ { n + 2 } ( x )$ using $I _ { n } ( x )$.
\end{enumerate}