Answer
$\text{The general solution satisfies the differential equation.}$
$\text{The particular solution is $y = -\ln 2 + \ln x$.}$
Work Step by Step
$\text{Let us find the second derivative of the general solution:}$
\begin{align}
& y = C_1 + C_2 \ln {x} \\
& y' = \frac{C_2}{x} \\
& y'' = -\frac{C_2}{x^2}
\end{align}
$\text{Then, substitute $y''$ and $y'$ into the differential equation}$
\begin{align}
& xy''+y'=0 \\
-\frac{C_2}{x^2} &\times x + \frac{C_2}{x} = 0 \\
& \ \ 0 = 0
\end{align}
$\text{Thus, the general solution satisfies the differential equation.}$
$\text{Now, we have to find the particular solution:}$
\begin{align}
& y = 0 \ \ when \ \ x =2 \\
& 0 = C_1 + C_2\ln 2 \\
& y' = \frac{1}{2} \ \ when \ \ x =2 \\
& \frac{1}{2} = \frac{C_2}{2} \Rrightarrow C_2 = 1 \Rrightarrow C_1 = -\ln 2
\end{align}
$\text{Thus, the particular solution is}$
\begin{align}
y = -\ln 2 + \ln x
\end{align}
