\textbf{Problem 1}

I. Find the general solution of the following differential equation:

$$\frac { d ^ { 4 } y } { d x ^ { 4 } } - 2 \cdot \frac { d ^ { 3 } y } { d x ^ { 3 } } + 2 \frac { d y } { d x } - y = 9 e ^ { - 2 x }$$

Here, $e$ denotes the base of the natural logarithm.

II. Find the value of the following integral:

$$\int _ { 0 } ^ { 1 } x ^ { m } ( \log x ) ^ { n } d x$$

Here, $m$ and $n$ are non-negative integers.

III. We define $I ( m )$ as

$$I ( m ) \equiv \int _ { 0 } ^ { 1 } x ^ { m } \arccos x \, d x$$

Here, $m$ is a non-negative integer. Use the principal values of inverse trigonometric functions.

\begin{enumerate}
  \item Find the value of $I ( 0 )$.
  \item Find the value of $I ( 1 )$.
  \item Express $I ( m )$ in terms of $m$ and $I ( m - 2 )$ when $m \geq 2$.
  \item Find the value of $I ( m )$.
\end{enumerate}