Question

The differential equation that represents the family of curves $y=\frac{1}{2x^2-c},$ where $c$ is an arbitrary constant is

• A. $\frac{dy}{dx}=4x{y}^2$
• B. $\frac{dy}{dx}=\frac{1}{y}$
• C. $\frac{dy}{dx}=x^{2}y$
• D. $\frac{dy}{dx}=-4x{y}^2$

Answer 1

The correct option is D $\frac{dy}{dx}=-4x{y}^2$

Given,
$y=\frac{1}{2x^2-c}$
$\Rightarrow 2x^2-c=\frac{1}{y}$
$\Rightarrow 2x^2-\frac{1}{y}=c$
Differentiating both sides, we get
$4x+\frac{1}{y^2}\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-4x{y}^2$