Question

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

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

Answer 1

Answer: (C)

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

$x^2+y^2-2ay=0$ (i)

Differentiating w.r.t $x$

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

$x+y{y}_1-2a{y}_1=0$ .....where $\frac{dy}{dx}=y_1$

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

Putting ${}^{\prime }{a}^{\prime }$ in (i)

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

$\Rightarrow (x^2-y^2)y^{\prime }=2xy$ which is required differential equation.