Question

The differential 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 2
• B. order 1, degree 1
• C. order 1, degree 3
• D. order 2, degree 2

Answer 1

The correct option is C order 1, degree 3

$y^2=2cx+{2c}^{3/2}$ .....(i)

$\Rightarrow 2y\frac{dy}{dx}=2c\Rightarrow y\frac{dy}{dx}=c$ ......(ii)
from (1) and (2)

$y^2=2y\frac{dy}{dx}\left(x\right)+2{\left(\frac{ydy}{dx}\right)}^{3/2}$

$\Rightarrow (y-2x\frac{dy}{dx})=2{\left(\frac{dy}{dx}\right)}^{3/2}{y}^{1/2}$
on squaring

$\Rightarrow {(y-2x\frac{dy}{dx})}^2=4{\left(\frac{dy}{dx}\right)}^3.y$

$\Rightarrow$ degree $=3$ and order $=1$