UPSC NDA I 2022 Mathematics Solved Paper

Concept:The equation $x \frac{dy}{dx} - 2y = 0$ is a first‑order variable‑separable differential equation. Its solution is obtained by separating variables, integrating, and simplifying to find the relation between $y$ and $x$.Explanation:Start with $x \frac{dy}{dx} - 2y = 0$.Rewrite as $\frac{dy}{dx} = \frac{2y}{x}$.Separate variables: $\frac{dy}{y} = \frac{2\,dx}{x}$.Integrate both sides: $\int \frac{dy}{y} = \int \frac{2\,dx}{x}$.The integrals give $\ln|y| = 2\ln|x| + \ln C$ (where $C$ is the constant of integration).Combine logs: $\ln|y| = \ln(C x^{2})$, so $y = C x^{2}$.The equation $y = C x^{2}$ represents for different $C$ values a set of vertical parabolas with vertex at the origin, opening upward or downward depending on the sign of $C$.Answer:Therefore, the given differential equation represents a family of parabolas, which corresponds to option C.