UPSC NDA I 2015 Mathematics Paper

Concept:The general solution of a first‑order differential equation is found by integration. The degree of a differential equation is the highest power of the highest‑order derivative after removing radicals.Explanation:For statement 1: $\frac{dy}{dx}=f(x)+x$.Integrating both sides: $y = \int \left( f(x)+x \right) dx = \int f(x)\,dx + \frac{x^{2}}{2} + c$.This is of the form $y = g(x) + c$, where $c$ is an arbitrary constant. Hence statement 1 is true.For statement 2: The equation $\left( \frac{dx}{dy} \right)^2 = f(x)$ is already a polynomial in $\frac{dx}{dy}$ with no radicals.The highest derivative is $\frac{dx}{dy}$ and its power is $2$. Therefore the degree is $2$. The transformation $\frac{dy}{dx} = \pm \frac{1}{\sqrt{f(x)}}$ is not needed to determine the degree; the given form directly gives degree $2$. Thus statement 2 is also true.Answer:Both statements 1 and 2 are correct. The correct option is C.