Question

The differential equation of family of curves $y^2=4a(x+a)$ is
a) $y^2=4\frac{dy}{dx}(x+\frac{dy}{dx})$
b) $2y\frac{dy}{dx}=4a$
c) $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$
d) $2x\frac{dy}{dx}+y{\left(\frac{dy}{dx}\right)}^2-y=0$

Answer 1

Given that, $y^2=4a(x+a)\dots \left(i\right)$
On differentiating both sides w.r.t. x, we get
$2y\frac{dy}{dx}=4a\Rightarrow 2y\frac{dy}{dx}=4a\Rightarrow y\frac{dy}{dx}=2a\Rightarrow a=\frac{1}{2}\frac{dy}{dx}\dots \left(ii\right)$
On putting the value of a from Eq. (ii) in Eq. (i), we get
$y^2=-2y\frac{dy}{dx}(x+\frac{1}{2}y\frac{dy}{dx})\Rightarrow y^2=2xy\frac{dy}{dx}+y^2{\left(\frac{dy}{dx}\right)}^2\Rightarrow 2x\frac{dy}{dx}+y{\left(\frac{dy}{dx}\right)}^2-y=0$

The option (d) is the required choice.