Question

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

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

Answer 1

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

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

$x^2+y^2-2ay=0$ ---------------------1

DIFFERENTIATE it

$2x+2y\frac{dy}{dx}-2a\frac{dy}{dx}=0$

$2x+(2y-2a)\frac{dy}{dx}=0$

$2x+2(y-a)y^{{}^{\prime }}=0$

$x=(a-y)y^{{}^{\prime }}=0$

$\frac{x}{y^{{}^{\prime }}}+y=a$ -------------------------------2

Put 2 in 1

$x^2+y^2-2y(\frac{x}{y^{{}^{\prime }}}+y)=0$

$x^2+y^2-2\frac{xy}{y^{{}^{\prime }}}-2y^2=0$

$x^2-y^2=\frac{2xy}{y^{{}^{\prime }}}$

$(x^2-y^2)y^{{}^{\prime }}=2xy$