Question

The differential equations satisfied by the system of parabolas $y^2=4a(x+a)$ is:

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

Answer 1

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

$y^2=4a(x+a)\cdots \left(1\right)$
$\Rightarrow 2y{y}^{\prime }=4a$
$\therefore y{y}^{\prime }=2a$
$\therefore by\left(1\right)y^2=2y{y}^{\prime }(x+\frac{y{y}^{\prime }}{2})$
$\Rightarrow y^2=2y{y}^{\prime }x+(y{y}^{\prime }{)}^2$
$\Rightarrow y(y^{\prime }{)}^2+2x{y}^{\prime }-y=0$
$($as $y\ne 0)$