Q. The differential equation obtained by eliminating the arbitrary constants $a$ and $b$ from $xy = ae^{x} + be^{- x}$ is
A) $x\ \frac{d^{2}y}{dx^{2}} + 2\frac{dy}{dx} - xy = 0$
B) $\frac{d^{2}y}{dx^{2}} + 2y\ \frac{dy}{dx} - xy = 0$
C) $x\ \frac{d^{2}y}{dx^{2}} + 2\ \frac{dy}{dx} + xy = 0$
D) $\frac{d^{2}y}{{dx}^{2}} + \frac{dy}{dx} - xy = 0$
Explanation
(a)
Given, $xy = ae^{x} + be^{- x}$
$\Rightarrow$ $x\ \frac{dy}{dx} + y = ae^{x} - be^{- x}$
$\Rightarrow$ $x\ \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} + \frac{dy}{dx} = ae^{x} + be^{- x}$
$\Rightarrow$ $x\ \frac{d^{2}y}{dx^{2}} + 2\ \frac{dy}{dx} - xy = 0$ [from eq. (I)]
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.