If the solution curve, of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _{\mathrm{e}}\left(\alpha+\left(\frac{y-1}{x-1}\right)^2\right)=\log _{\mathrm{e}}|x-1|$, then $5 \beta+\alpha$ is equal to __________.