The differential equation representing the family of curves $y^2=2 c(x+\sqrt{c})$, where c is a positive parameter, is of

order 1 and degree 3

We have,

$y^2=2 c(x+\sqrt{c})$            .......(i)

$\Rightarrow 2 y y_1=2 c$

$\Rightarrow y y_1=c$           .......(ii)

Eliminating c from (i) and (ii), we get

$y^2=2 y y_1\left(x+\sqrt{y y_1}\right)$

$\Rightarrow y-2 x y_1=\left(\sqrt{y} y_1\right)^{3 / 2} \Rightarrow\left(y-2 x y_1\right)^2=\left(y y_1\right)^3$

Clearly, it is a differential equation of order one and degree 3.