UPSC NDA I 2023 Mathematics Solved Paper

Concept:The degree of a differential equation is the highest power of the highest order derivative present, after the equation is made free from radicals and fractions in the derivatives.Explanation:Start with the given equation: $y^2 = 2cx + 2c\sqrt{c}$.Differentiate both sides with respect to $x$: $2yy' = 2c$ (since $\frac{d}{dx}(c) = 0$).Simplify to get $c = yy'$.Substitute $c = yy'$ back into the original equation: $y^2 = 2(yy')x + 2(yy')^{3/2}$.Rearrange to isolate the radical term: $y^2 - 2xyy' = 2(yy')^{3/2}$.Square both sides to remove the square root: $(y^2 - 2xyy')^2 = 4(yy')^3$.Now the equation is a polynomial in $y'$ (the first derivative). The highest power of $y'$ is $3$ on the right-hand side.Since the highest order derivative is $y'$ (order 1), and its highest exponent is 3, the degree of the differential equation is 3.Answer:3