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 $x ( 0 ) = \frac { 1 } { 2 }$.
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 $x ( 0 ) = \frac { 1 } { 2 }$.