QJEEM24Maf18f: JEE Main 2024 Mathematics - Class 12 JEEM24Maf18f Solution: Let be the solution of the differential equation . Then is…

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

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