Question
The differential equation for the family of circle $${x^2} + {y^2} - 2ay = 0,$$ where $$a$$ is an arbitrary constant is -
A.
$$\left( {{x^2} + {y^2}} \right)y' = 2xy$$
B.
$$2\left( {{x^2} + {y^2}} \right)y' = xy$$
C.
$$\left( {{x^2} - {y^2}} \right)y' = 2xy$$
D.
$$2\left( {{x^2} - {y^2}} \right)y' = xy$$
Answer :
$$\left( {{x^2} - {y^2}} \right)y' = 2xy$$
Solution :
$$\eqalign{
& {x^2} + {y^2} - 2ay = 0\,.....\left( 1 \right) \cr
& {\text{Differentiate,}} \cr
& 2x + 2y\frac{{dy}}{{dx}} - 2a\frac{{dy}}{{dx}} = 0\,\, \Rightarrow a = \frac{{x + yy'}}{{y'}} \cr
& {\text{Put in }}\left( 1 \right){\text{,}}\,\,{x^2} + {y^2} - 2\left( {\frac{{x + yy'}}{{y'}}} \right)y = 0 \cr
& \Rightarrow \left( {{x^2} + {y^2}} \right)y' - 2xy - 2{y^2}y' = 0 \cr
& \Rightarrow \left( {{x^2} - {y^2}} \right)y' = 2xy \cr} $$