I. Answer the following questions about the differential equation:

$$\cos x \frac { d ^ { 2 } y } { d x ^ { 2 } } - \sin x \frac { d y } { d x } - \frac { y } { \cos x } = 0 \quad \left( - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \right) .$$

\begin{enumerate}
  \item A particular solution of Eq. (1) is of the form of $y = ( \cos x ) ^ { m }$ (m is a constant). Find the constant $m$.
  \item Find the general solution of Eq. (1), using the solution of Question I.1.
\end{enumerate}

II. Find the value of the following integral:

$$I = \int _ { 1 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 4 } + 2 x ^ { 2 } - 1 } d x$$

Note that, for a positive constant $\alpha$, the relation $\int _ { 0 } ^ { \infty } e ^ { - \alpha x ^ { 2 } } d x = \frac { 1 } { 2 } \sqrt { \frac { \pi } { \alpha } }$ holds.

III. Express the general solution of the following differential equation in the form of $f ( x , y ) = C$ ( $C$ is a constant) using an appropriate function $f ( x , y )$ :

$$\left( x ^ { 3 } y ^ { n } + x \right) \frac { d y } { d x } + 2 y = 0 \quad ( x > 0 , y > 0 )$$

where $n$ is an arbitrary real constant.