$$\begin{aligned} & (2 x \log x) \frac{d y}{d x}+2 y=\frac{3}{x} \log x \\ & \frac{d y}{d x}+\frac{y}{x \log x}=\frac{3}{2 x^2} \\ & I F=e^{\int \frac{1}{x \log x}}=e^{\log (\log x)}=\log x \\ & y \times \log x=\int \log x \times \frac{3}{2 x^2} d x+C \\ & y \log x=\frac{3}{2}\left[\frac{-\log x}{x}-\frac{1}{x}\right]+C \\ & y\left(e^{-1}\right)=0 \\ & \Rightarrow 0=\frac{3}{2}[e-e]+C \\ & C=0 \\ & y \log x=\frac{-3}{2}\left[\frac{\log x+1}{x}\right] \\ & y(e) \rightarrow y(e) \times 1=\frac{-3}{2}\left[\frac{1+1}{e}\right]=-\frac{3}{e} \end{aligned}$$