Q. A solution of the differential equation $\left( \ \frac{dy}{dx}\ \right)^{2} - x\ \frac{dy}{dx} + y = 0$ is
A) $y = 2$
B) $y = 2x$
C) $y = 2x - 4$
D) $y = 2x^{2} - 4$
Explanation
(c)
Let $p = \frac{dy}{dx}$
$\therefore$ Given differential equation reduces to
$p^{2} - xp + y = 0$
Differentiating both sides $w.r.t.x$, we get
$2p\frac{dp}{dx} - x\frac{dp}{dx} - p + p = 0$
$\Rightarrow \frac{dp}{dx}(2p - x) = 0$
$\Rightarrow$ Either $\frac{d^{2}y}{dx^{2}} = 0$ or $\frac{dy}{dx} = \frac{x}{2}$
$\Rightarrow y = 2x - 4$ will satisfy.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.