UPSC NDA 2 2022 Mathematics Solved Paper

Concept:Order refers to the highest derivative present in the equation. Degree is the power of the highest derivative, but it is strictly defined only when the differential equation can be expressed as a polynomial in the derivatives.Explanation:Let us analyze Statement 1: The differential equation is $\frac{dy}{dx} + \cos\left(\frac{dy}{dx}\right) = 0$. This equation involves a cosine function of the derivative $\frac{dy}{dx}$. For the degree to be defined, the equation must be a polynomial in the derivatives. Since $\cos\left(\frac{dy}{dx}\right)$ is a transcendental term, the equation is not a polynomial in the derivative. Therefore, the degree of this differential equation is not defined. Statement 1, which claims the degree is 1, is incorrect. Let us analyze Statement 2: The differential equation is $\left(\frac{d^2y}{dx^2}\right)^3 + \cos\left(\frac{dy}{dx}\right) = 0$. The order of a differential equation is determined solely by the highest derivative present in the equation. Here, the highest derivative is $\frac{d^2y}{dx^2}$, which is of order 2. This holds true irrespective of whether the equation is a polynomial or contains non-polynomial terms. Therefore, the order is exactly 2. Statement 2 is correct.Answer:Only Statement 2 is correct. Hence, the correct option is B (2 only).