I. Answer the following questions about the differential equation:
$$\cos x \frac { d ^ { 2 } y } { d x ^ { 2 } } - \sin x \frac { d y } { d x } - \frac { y } { \cos x } = 0 \quad \left( - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \right) .$$

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

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.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Show that the value of $|f(x)|^{2} + |g(x)|^{2}$ is independent of $x$, where $|A|$ denotes the absolute value of a complex number $A$.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Let $a = 0.8$ and $b = 0.6$. Solve the simultaneous ordinary differential equations derived in Question II.1 using the initial values $f(0) = 1$ and $g(0) = 0$, and obtain $f(x)$ and $g(x)$.

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.