UPSC NDA 2 2019 Mathematics Solved Paper

Concept:The degree of a differential equation is the power of the highest‑order derivative present in it, after the equation is made polynomial in derivatives.Explanation:The given differential equation is:$\frac{d^{3}y}{dx^{3}} + \left(\frac{dy}{dx}\right)^{2} - x^{2}\frac{d^{4}y}{dx^{4}} = 0$First, identify the highest‑order derivative. Here, the term $\frac{d^{4}y}{dx^{4}}$ is of order 4, which is the highest.Now, observe the power of this highest‑order derivative. It appears as $x^{2}\frac{d^{4}y}{dx^{4}}$, with no exponent shown beyond 1. Thus, its power is 1.There are no radicals or fractional powers involving the derivatives, and the equation is already a polynomial in the derivatives.Hence, the degree is simply the exponent of $\frac{d^{4}y}{dx^{4}}$, which is 1.Answer:The degree of the differential equation is 1. (Option A)