Let $t$ be a real independent variable, and let $x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Find all solutions of the following ordinary differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = \cos ( t )$$
which are bounded when $t \rightarrow - \infty$.
(2) Find all solutions $x ( t )$ and $y ( t )$ of the following ordinary differential equations
$$\begin{aligned} & \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x - y = \cos ( t ) \\ & \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + y - x = 0 \end{aligned}$$
which are bounded when $t \rightarrow - \infty$.
(3) By converting the following ordinary differential equation
$$e ^ { - t } x ^ { 2 } - 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 0$$
to a linear ordinary differential equation with an appropriate change of variable, find the solution $x ( t )$ that satisfies $x ( 0 ) = \frac { 1 } { 2 }$.

Problem 1

I. Find the general solutions $y ( x )$ for the following differential equations:

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

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

1. Calculate the area of the region enclosed by the curve $C$.
2. 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.