\section*{Problem 1}
I. Find the value of the following definite integral:

$$I = \int _ { 2 } ^ { 4 } \frac { d x } { \sqrt { ( x - 2 ) ( 4 - x ) } }$$

II. Find the general solution and the singular solution of the following differential equation:

$$y = x \frac { d y } { d x } + \frac { d y } { d x } + \left( \frac { d y } { d x } \right) ^ { 2 }$$

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

$$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } - 8 y = x ^ { 2 }$$