Question

The differential equation whose solution represents the family $xy=A{e}^{ax}+B{e}^{-ax}$ is

• A. $x{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+2\frac{dy}{dx}=xy$
• B. $x{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+2\frac{dy}{dx}=a^{2}xy$
• C. $x\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}=a^{2}xy$
• D. none of these

Answer 1

Answer: (C)

Given $xy=A{e}^{ax}+B{e}^{-ax}\dots (i)$
Differentiating $(i)$ w.r.t. $x$, we get
$y+x\frac{dy}{dx}=a\left(A{e}^{ax}-B{e}^{-ax}\right)$
Differentiating again w.r.t. $x$, we get
$\frac{dy}{dx}+x\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=a^2\left(A{e}^{ax}+B{e}^{-ax}\right)$
$\Rightarrow x\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}=a^2\left(xy\right)$ (using $(i)$)