I. Find the general solutions of the following differential equations.

\begin{enumerate}
  \item $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = e ^ { x } \cos x$
  \item $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { 1 } { x } \frac { d y } { d x } + \frac { 4 } { x ^ { 2 } } y = \left( \frac { 2 \log x } { x } \right) ^ { 2 }$
\end{enumerate}

II. Answer the following questions for the partial differential equation represented in Equation (3) and the boundary conditions represented in Equations (4)-(7):

$$\begin{aligned}
& \frac { \partial ^ { 2 } u ( x , y ) } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u ( x , y ) } { \partial y ^ { 2 } } = 0 \quad ( 0 \leq x , 0 \leq y \leq 1 ) \\
& \left\{ \begin{array} { l } 
\lim _ { x \rightarrow + \infty } u ( x , y ) = 0 \\
\left. \frac { \partial u ( x , y ) } { \partial y } \right| _ { y = 0 } = 0 \\
u ( x , 1 ) = 0 \\
\left. \frac { \partial u ( x , y ) } { \partial x } \right| _ { x = 0 } = 1 + \cos \pi y
\end{array} \right.
\end{aligned}$$

\begin{enumerate}
  \item Find the solution which satisfies Equations (3) and (4) in the form of $u ( x , y ) = X ( x ) \cdot Y ( y )$.
  \item Find the solution satisfying Equations (5) and (6) for the solution of Question II.1.
  \item Find the solution of the partial differential equation (3) satisfying all the boundary conditions given in Equations (4)-(7), using the solution of Question II.2.
\end{enumerate}