UPSC NDA I 2022 Mathematics Solved Paper

Concept:The degree of a differential equation is the power of the highest order derivative, after the equation is made free from radicals and fractions.Explanation:The given equation is $1 + \left( \frac{dy}{dx} \right)^2 = \left( \frac{d^2y}{dx^2} \right)^{\frac{4}{3}}$.The highest order derivative is $\frac{d^2y}{dx^2}$.Its exponent is a fraction $\frac{4}{3}$.To remove the fractional exponent, cube both sides:$\left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = \left[ \left( \frac{d^2y}{dx^2} \right)^{\frac{4}{3}} \right]^3$.Using $(a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2$, the left side expands to:$1 + \left( \frac{dy}{dx} \right)^6 + 3\left( \frac{dy}{dx} \right)^2 + 3\left( \frac{dy}{dx} \right)^4$.The right side simplifies using $(a^m)^n = a^{mn}$ to $\left( \frac{d^2y}{dx^2} \right)^{4}$.Thus the equation becomes:$1 + \left( \frac{dy}{dx} \right)^6 + 3\left( \frac{dy}{dx} \right)^2 + 3\left( \frac{dy}{dx} \right)^4 = \left( \frac{d^2y}{dx^2} \right)^4$.Now the highest order derivative $\frac{d^2y}{dx^2}$ has an exponent of 4.Therefore, the degree of the differential equation is 4.Answer:D. 4