COMEDK 2025 Morning Solved Paper

Concept:The order of a differential equation is the highest order derivative present after simplifying the equation. Here, the given expression involves a derivative of a cubed first derivative.Explanation:Start with the expression: $\frac{d}{dx}\left[ \left(\frac{dy}{dx}\right)^3 \right] = 0$.Let $p = \frac{dy}{dx}$. Then the left side becomes $\frac{d}{dx}(p^3)$.Using the chain rule, $\frac{d}{dx}(p^3) = 3p^2 \cdot \frac{dp}{dx}$.But $\frac{dp}{dx} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{d^2 y}{dx^2}$, which is the second derivative of $y$ with respect to $x$.Thus, the equation simplifies to $3\left(\frac{dy}{dx}\right)^2 \cdot \frac{d^2 y}{dx^2} = 0$.The highest derivative that appears in this simplified equation is the second derivative $\frac{d^2 y}{dx^2}$.Therefore, the order of the differential equation is 2.Answer:Option C (2).