Question

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

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

Answer 1

Answer: (A)

Given, $xy=ae^{x}+be^{- x}$ ..... (I)
$\Rightarrow $ $x \, \frac{d y}{d x}+y=ae^{x}-be^{- x}$
$\Rightarrow $ $x \, \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+\frac{d y}{d x}=ae^{x}+be^{- x}$
$\Rightarrow $ $x \, \frac{d^{2} y}{d x^{2}}+2 \, \frac{d y}{d x}-xy=0$ [from eq. (I)]