Question

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