$$\begin{aligned} & \alpha|x|=|y| e^{x y-\beta} \\ & \frac{x d y-y d x}{y^2}+\frac{x y(x d y+y d x)}{y^2}=0 \\ & -d\left(\frac{x}{y}\right)+\frac{x}{y} d(x y)=0 \end{aligned}$$

$$\begin{aligned} & \int d(x y)=\int \frac{d\left(\frac{x}{y}\right)}{\frac{x}{y}} \\ & x y=\ln \left|\frac{x}{y}\right|+\ln c \\ & x y=\ln \left(\left|\frac{x}{y}\right| \cdot c\right) \\ & \because y(1)=2 \\ & 2=\ln \left|\frac{1}{2}\right| c \Rightarrow c=2 e^2 \\ & \therefore \quad \operatorname{solution} x y=\ln \left(\left|\frac{x}{y}\right| \cdot 2 e^2\right) \\ & e^{x y}=\frac{|x|}{|y|} \cdot 2 e^2 \\ & 2|x|=|y| e^{x y-2} \\ & \Rightarrow \alpha=2, \beta=2, \alpha+\beta=4 \end{aligned}$$