Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
Solution of the differential equation $x^2\left(y - x\frac{dy}{dx}\right) = y\left(\frac{dy}{dx}\right)^2$ which does not contain singular solution is :
$x^2(y - xc) = yc^2$
$y = cx + c^2$
$y^2 = cx^2 + c^2$
$xy = cx^2 + c$
Step-by-Step Solution
Key Concept: Substituting $x^2 = u$ and $y^2 = v$ transforms the equation into Clairaut's standard form.
Substitute $x^2 = u$ and $y^2 = v$ to transform the differential equation. This gives $\frac{dy}{dx} = \frac{x}{y}\left(\frac{dv}{du}\right)$. The resulting equation becomes Clairaut's equation in variables $u$ and $v$: $v = uc + c^2$ or $y^2 = cx^2 + c^2$, which is the general solution.
Correct Answer: 3