Question

The differential equation for the family of curves $x^2+y^2-2ay=0$, where ${}^{\prime }{a}^{\prime }$ is an arbitrary constant is

• A. $(x^2-y^2)y^{\prime }=2xy$
• B. $2(x^2+y^2)y^{\prime }=xy$
• C. $2(x^2-y^2)y^{\prime }=xy$
• D. $(x^2+y^2)y^{\prime }=2xy$

Answer 1

The correct option is A $(x^2-y^2)y^{\prime }=2xy$

Given family of curve is
$x^2+y^2-2ay=0$ ....... $\left(1\right)$
$\Rightarrow$ $2a=\frac{x^2+y^2}{y}$
Also from $\left(1\right)$,

$2x+2y{y}^{\prime }-2a{y}^{\prime }=0$
$\Rightarrow$ $2x+2y{y}^{\prime }-\left(\frac{x^2+y^2}{y}\right)y^{\prime }=0$
$\Rightarrow$ $2xy+y^{\prime }(2y^2-x^2-y^2)=0$
$\Rightarrow$ $y^{\prime }(x^2-y^2)=2xy$