UPSC NDA 2 2021 Mathematics Solved Paper

Concept:To form a differential equation from a given relation, we differentiate successively to eliminate the arbitrary constants (here $A$ and $B$).Explanation:Start with $y = A - \frac{B}{x}$.Differentiate once with respect to $x$: $\frac{dy}{dx} = y_1 = \frac{B}{x^2}$.Thus $x^2 y_1 = B$.Differentiate again: $\frac{d}{dx}(x^2 y_1) = \frac{d}{dx}(B) = 0$.Using product rule: $2x y_1 + x^2 y_2 = 0$.Factor out $x$: $x(2y_1 + x y_2) = 0$.Since $x \neq 0$ in general, we get $x y_2 + 2 y_1 = 0$.Answer:The required differential equation is $x y_2 + 2 y_1 = 0$, which matches option B.