UPSC NDA I 2023 Mathematics Solved Paper

Concept: The order of a differential equation is the highest order derivative present after eliminating arbitrary constants.Explanation:Given relation: $y^{2} = 2c x + 2c\sqrt{c}$, where $c$ is an arbitrary constant.Differentiate both sides with respect to $x$: $2y y' = 2c$ ⇒ $c = y y'$.Substitute $c$ back into the original relation: $y^{2} = 2(y y') x + 2(y y')^{3/2}$.Simplify: $y^{2} - 2x y y' = 2 (y y')^{3/2}$.Square both sides to remove the fractional exponent: $(y^{2} - 2x y y')^{2} = 4 (y y')^{3}$.The highest order derivative in this differential equation is $y'$ (first derivative).Therefore, the order is $1$.Answer: Option A: $1$.