Question

The differential equation satisfied by the curves $e^{2y}+2bx{e}^y+b^2=0,$ where $b$ is a parameter, is

• A. $(1+x^2)\frac{d^{2}y}{d{x}^2}=0$
• B. $(x^2-1){\left(\frac{dy}{dx}\right)}^2-1=0$
• C. $(x^2-1)\frac{d^{2}y}{d{x}^2}+2=0$
• D. $(x^2-x)\frac{dy}{dx}-y^2=0$

Answer 1

The correct option is B $(x^2-1){\left(\frac{dy}{dx}\right)}^2-1=0$

$e^{2y}+2bx{e}^y+b^2=0$
$\Rightarrow e^{2y}+2bx{e}^y+b^2{x}^2=b^2{x}^2-b^2$
$\Rightarrow (e^y+bx{)}^2=b^2(x^2-1)$
$\Rightarrow e^y+bx=\pm b\sqrt{x^2-1}$
$\Rightarrow e^y=-b(x\pm \sqrt{x^2-1})\cdots \left(1\right)$

Differentiating equation $\left(1\right)$ with respect to $x,$ we get
$e^y\frac{dy}{dx}=\frac{-b(\sqrt{x^2-1}\pm x)}{\sqrt{x^2-1}}\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right),$
$\sqrt{x^2-1}\frac{dy}{dx}=\pm 1$
$\Rightarrow (x^2-1){\left(\frac{dy}{dx}\right)}^2=1$