Question

The differential equation obtained by eliminating the arbitrary constants $a$ and $b$ from $xy=a{e}^x+b{e}^{-x}$ is

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

Answer 1

Answer: (A)

Given, $xy=a{e}^x+b{e}^{-x}$ ..... (I)
$\Rightarrow$ $x\frac{dy}{dx}+y=a{e}^x-b{e}^{-x}$
$\Rightarrow$ $x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+\frac{dy}{dx}=a{e}^x+b{e}^{-x}$
$\Rightarrow$ $x\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-xy=0$ [from eq. (I)]