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

Order 1 and degree 3

The correct answer is option (1) : Order 1 and degree 3

We have,

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

$⇒2y\, y _1= 2c$

$⇒y y_1= c $ .............(ii)

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

$y^2 = 2y\, y_1(x+ \sqrt{yy_1})$

$⇒y - 2xy_1= (\sqrt{u}y_1)^{3/2}⇒(y-2xy_1)^2=(yy_1)^3$

Clearly, it is a differential equation of order 1 and degree 3.