Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade None
Question:
If the independent variable $x$ is changed to $y$, then the differential equation $x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 - \frac{dy}{dx} = 0$ is changed to $x\frac{d^2x}{dy^2} + \left(\frac{dx}{dy}\right)^2 = k$ where $k$ equals ______.
Step-by-Step Solution
Key Concept: Use the substitution $p = \frac{dy}{dx}$ and interchange variables to reduce the order and simplify the differential equation.
Given $x\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - \frac{dy}{dx} = 0$, use the substitution $p = \frac{dy}{dx}$ so that $\frac{d^2y}{dx^2} = p\frac{dp}{dy}$. This transforms the equation to $xp\frac{dp}{dy} + p^2 - p = 0$. Rearranging and integrating yields $x\frac{d^2x}{dy^2} + \left(\frac{dx}{dy}\right)^2 = 1$ after switching variables.
Correct Answer: 1