\section*{Problem 1}
I. Find the general solutions $y ( x )$ for the following differential equations:

\begin{enumerate}
  \item $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \left( \frac { y } { x } \right) ^ { 3 }$,
  \item $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 [ y + \cos ( 3 x ) ] = 0$.
\end{enumerate}

II. Consider the curve $C$ given by the following polar equation in the polar coordinate system $( r , \theta )$ with the origin $O$ on the $x y$-orthogonal coordinate plane as the pole, and the positive part of the $x$-axis as the starting line:

$$r = 2 + \cos \theta \quad ( 0 \leq \theta < 2 \pi )$$

\begin{enumerate}
  \item Calculate the area of the region enclosed by the curve $C$.
  \item Consider the tangent line at the point $( r , \theta ) = \left( \frac { 4 + \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right)$ on the curve $C$. Find the slope of this tangent line in the $x y$-orthogonal coordinate system.
\end{enumerate}