Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
The solution of $y = 2x\left(\frac{dy}{dx}\right) + x^2\left(\frac{dy}{dx}\right)^4$ is :
$y = 2cx^{1/2}y^{1/4} + c$
$y = 2\sqrt{cx^2 + c^2}$
$y = 2\sqrt{c(x+1)}$
$y = 2\sqrt{cx + c^2}$
Step-by-Step Solution
Key Concept: Separating variables after recognizing the differential form yields a tractable equation in $p$ and $x$.
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