Q. The differential equation for the family of curves $x^{2} + y^{2} - 2ay = 0$, where $a$ is an arbitrary constant, is
A) $2\left( x^{2} - y^{2} \right)y^{'} = xy$
B) $2\left( x^{2} + y^{2} \right)y^{'} = xy$
C) $\left( x^{2} - y^{2} \right)y^{'} = 2xy$
D) $\left( x^{2} + y^{2} \right)y^{'} = 2xy$
Explanation
(c)
Given, $x^{2} + y^{2} - 2ay = 0$ ….(i)
$\Rightarrow$ $2x + 2yy^{'} - 2ay^{'} = 0$
$\Rightarrow$ $\frac{2x + 2yy'}{y'} = 2a$ …(ii)
$\therefore$ From Eq. (i)
$2a = \frac{x^{2} + y^{2}}{y}$
$\Rightarrow$ $\frac{2x + 2yy'}{y'} = \frac{x^{2} + y^{2}}{y}$ [from Eq. (ii)]
$\Rightarrow$ $\left( x^{2} - y^{2} \right)y^{'} = 2xy$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.