Question

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

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

Answer 1

Answer: (B)

Key Idea : The differential equation of a family of curves of $n$ parameters is a differential equation of $n$ maximum order.
Equation of family of curves is
$y^2=2c\left(x+\sqrt{c}\right)...\left(i\right)$
On differentiating Eq. $\left(i\right)$ with respect to $x$, then
$2y{y}_1=2c$
$\Rightarrow c=y{y}_1$
On putting the value of $c$ in Eq. $\left(i\right)$, we get
$y^2=2y{y}_1\left(x+\sqrt{y{y}_1}\right)$
$\Rightarrow \left(y^2-2y{y}_1{x}^2\right)=4{\left(y{y}_1\right)}^3$
$\therefore$ The degree and order of above equation are 3 and 1 respectively.