9. differential equations — ગણિત ધોરણ 12 સાયન્સ — Question

$\Rightarrow \frac{d y}{e^{y}}=e^{x} d x$

$\Rightarrow \mathrm{e}^{-\mathrm{y}} \mathrm{dy}=\mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}$

Intergrating both sides, we get:

$\int e^{-y} d y=\int e^{x} d x$

$\Rightarrow-e^{-y}=e^{x}+k$

$\Rightarrow \mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{y}}=-\mathrm{k}$

$\Rightarrow e^{x}+e^{-y}=c \quad(c=-k)$

Hence, the correct answer is $\mathrm{C}.$