Variable Separable

The exponential e^{x-y} factors as e^x \cdot e^{-y}, which immediately reveals a separable structure because the x-dependence and y-dependence split as a product. The strategic step is to recognize that any exponential of a sum or difference of variables decomposes into separable factors. Rewriting the equation, \frac{dy}{dx} = e^x e^{-y}, and separating gives e^{y},dy = e^{x},dx. Integrating both sides yields e^{y} = e^{x} + C, where C is the single arbitrary constant. Option e^{-y} = e^x + C arises from forgetting to move the negative exponential to the left correctly, which would flip the sign of the y exponent. Option y = e^x + C omits the exponential on the left entirely, treating dy as though it integrated to y rather than e^y,dy integrating to e^y. Option e^y = e^{-x} + C reverses the sign of the x exponent and contradicts the separation. This is the textbook JEE Advanced separable-exponential pattern. As a final plausibility check, differentiating e^y = e^x + C implicitly gives e^y y' = e^x, so y' = e^x e^{-y} = e^{x-y}, exactly the original equation, confirming the solution is correct.