UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The order of a differential equation is the highest order of derivative present in the equation after eliminating arbitrary constants.Explanation:The standard equation of an ellipse with axes along the coordinate axes is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, where $a$ and $b$ are arbitrary constants.Differentiate once with respect to $x$: $\frac{2x}{a^{2}} + \frac{2y y'}{b^{2}} = 0$, which simplifies to $\frac{x}{a^{2}} + \frac{y y'}{b^{2}} = 0$. (Equation 1)Differentiate again with respect to $x$: $\frac{1}{a^{2}} + \frac{y y'' + (y')^{2}}{b^{2}} = 0$. (Equation 2)From these two equations, eliminate $a$ and $b$ to obtain a differential equation free of arbitrary constants.Solving gives: $x y y'' + (y')^{2} - y y' = 0$.The highest derivative present is $y''$ (second order).Answer:Order = 2, corresponding to option B.