Q. The differential equation obtained by eliminating arbitrary constants from $y = ae^{bx}$ is
A) $y\ \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} = 0$
B) $y\ \frac{d^{2}y}{dx^{2}} - \frac{dy}{dx} = 0$
C) $y\ \frac{d^{2}y}{dx^{2}} - \left( \frac{dy}{dx} \right)^{2} = 0$
D) $y\ \frac{d^{2}y}{dx^{2}} + \left( \frac{dy}{dx} \right)^{2} = 0$
Explanation
(c)
The given equation is
$y = ae^{bx}$
$\Rightarrow$ $\frac{dy}{dx} = abe^{bx}$ …(i)
$\Rightarrow$ $\frac{d^{2}y}{dx^{2}} = ab^{2}e^{bx}$ …(ii)
$\Rightarrow$ $ae^{bx}\ \ \frac{d^{2}y}{dx^{2}} = a^{2}b^{2}e^{2bx}$
$\Rightarrow$ $y\ \ \frac{d^{2}y}{dx^{2}} = \left( \frac{dy}{dx} \right)^{2}$ [from eq. (ii)]
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.