Question

The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c>0$, is a parameter, is of order and degree as follows:

• A. order $1$, degree $1$
• B. order $1$, degree $2$
• C. order $2$, degree $2$
• D. order $1$, degree $3$

Answer 1

The correct option is D order $1$, degree $3$

Given equation is $y^2=2c(x+\sqrt{c})$ ......(1)
$\therefore$ $2y{y}_1=2c$
$\therefore$ $y{y}_1=c$
Now putting $c=y{y}_1$ in (1), we get
$y_2=2.y{y}_1(x+\sqrt{y{y}_1})$

$\Rightarrow$ ${(y^2-2xy{y}_1)}^2=4{\left(y{y}_1\right)}^3$
$\Rightarrow$ ${(y^2-2xy{y}_1)}^2=4y^3{y}_1^3$

$\Rightarrow$ order $1$, degree $3$