UPSC NDA 2 2016 Mathematics Solved Paper

Concept:The order of a differential equation is the order of the highest derivative present in it. If an equation contains an integral operator, it must be converted to a purely differential form before determining the order.Explanation:The given equation is: $\frac{dx}{dy} + \int y \, dx = x^3$.Here, $\int y \, dx$ is an integral operator, not a derivative.To remove the integral, differentiate both sides with respect to $x$.This gives: $\frac{d}{dx}\left(\frac{dx}{dy}\right) + y = 3x^2$.The term $\frac{d}{dx}\left(\frac{dx}{dy}\right)$ involves a derivative of $\frac{dx}{dy}$ with respect to $x$.Using the chain rule, $\frac{d}{dx}\left(\frac{dx}{dy}\right) = \frac{d^2x}{dy^2} \cdot \frac{dy}{dx}$.This expression contains a second derivative $\frac{d^2x}{dy^2}$, but it is mixed with $\frac{dy}{dx}$ and cannot be simplified into a standard derivative form like $\frac{dy}{dx}$ or $\frac{d^2y}{dx^2}$ without more information about the relationship between $x$ and $y$.Therefore, the given equation cannot be expressed as a standard differential equation with a clear order.Answer:The order of the given differential equation cannot be determined.