The solution of $y = 2x\left(\frac{dy}{dx}\right) + x^2\left(\frac{dy}{dx}\right)^4$ is :
Step-by-Step Solution
Key Concept: Recognize this as a Clairaut-type differential equation. Substitute p = dy/dx and differentiate the original equation with respect to x, then separate variables to obtain dp/p = -dx/x, which yields p²x = c.
Write $p = \frac{dy}{dx}$ and differentiate to get $p + 2x\frac{dp}{dx} + 2xp^3 + 4p^3x = 0$. This simplifies to $\frac{dp}{p} = -\frac{dx}{x}$, giving $2\log p + \log x = \text{const}$. Therefore $p^2x = c$ or $p = \sqrt{\frac{c}{x}}$, yielding option 'D'.
Correct Answer: 4