Differential Equations

If the solution curve \(y=y(x)\) of the differential equation \(\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0\) passes through the point \((1,1)\) and \(y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan \left(\frac{3}{2}\right)}\), then \(\alpha+2 \beta\) is _________.