Question

Find the differential equation of the family of curves given by $x^2+y^2=2ay$.

Answer 1

Given,

$\begin{array}{c}{x}^2+y^2=2ay\\ x^2+y^2-2ay=0\end{array}$

Differentiating w.r.t $x$,

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

Divide throughout by $2$,

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

$a=\frac{x+y\left(\frac{dy}{dx}\right)}{d\frac{dy}{dx}}$

Substitute the value of a in the given equation,

$\begin{array}{c}{x}^2+y^2=2x\frac{x+y\left(\frac{dy}{dx}\right)}{\frac{dy}{dx}}\\ \frac{dy}{dx}\left(x^2+y^2\right)=-2xy\end{array}$