Question

The differential equation obtained by eliminating arbitrary constants from $y=a{e}^{bx}$ is

• A. $y\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=0$
• B. $\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$
• C. $y\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$
• D. $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$

Answer 1

Answer: (C)

The given equation is $y=a{e}^{bx}$
$\Rightarrow \frac{dy}{dx}=ab{e}^{bx}$
$\Rightarrow \frac{d^{2}y}{d{x}^2}=a{b}^2{e}^{bx}\dots$(i)
$\Rightarrow a{e}^{bx}\frac{d^{2}y}{d{x}^2}=a^2{b}^2{e}^{bx}\dots$(ii)
$\Rightarrow y\frac{d^{2}y}{d{x}^2}={\left(\frac{dy}{dx}\right)}^2[$ from eq. (ii) $]$
$\Rightarrow y\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$