\textbf{Problem 1}

\textbf{I.} Obtain the general solution of the following differential equation:
$$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } + y = x ^ { 3 }$$

\textbf{II.} Obtain the general solution of the following differential equation:
$$x ^ { 2 } \frac { d y } { d x } - x ^ { 2 } y ^ { 2 } + x y + 1 = 0$$
Note that $y = \frac { 1 } { x }$ is a particular solution.

\textbf{III.} Let $I _ { n }$ be defined by:
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { n } x \, d x$$
where $n$ is a non-negative integer.
\begin{enumerate}
  \item Calculate $I _ { 0 } , I _ { 1 }$, and $I _ { 2 }$.
  \item Calculate $I _ { n }$ for $n \geq 2$.
\end{enumerate}