Given that \( y_{1}=e^{x} \) is a solution of \( x y^{\prime \prime}+(1-2 x) y^{\prime}+(x-1) y=0 \), find its second solution \( y_{2} \) on \( (0, \infty) \), which is linearly independent from \( y_{1} \).
\( x e^{x} \)
\( (x-1) e^{x} \)
\( e^{-2 x} \ln x \)
\( e^{-x} \ln x \)
\( e^{x} \ln x \)
