Solve: \( y^{\prime}+\frac{3}{x} y=g(x) \), where \( g(x)=\left\{\begin{array}{l}1 \text { if } ...

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Solve:
\( y^{\prime}+\frac{3}{x} y=g(x) \), where \( g(x)=\left\{\begin{array}{l}1 \text { if } 0 \leq x \leq 1 \\ \frac{1}{x} \text { if } x1\end{array}\right. \) and \( y\left(\frac{1}{2}\right)=\frac{1}{8} \), and \( y(x) \) is continuous on \( [0, \infty) \).
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