Question

$y{\left(\frac{dy}{dx}\right)}^2+y=2xy\frac{dy}{dx}$

Answer 1

Given,

$y{\left(\frac{dy}{dx}\right)}^2+y=2xy\frac{dy}{dx}$

substitute $\frac{dy}{dx}=p$

$y{p}^2+y=2xyp$

$\Rightarrow y=2xy-2yp$.....(1)

using Clairaut's equation, differentiate both sides of (1) w.r.t p

$\frac{dy}{dp}=2xy-2yp$

To find the solution, set $\frac{dy}{dp}=0$

$0=2xy-2yp$

$\therefore p=x$

substitute the value of p in equation (1)

$y=2x^{2}y-y{x}^2$

$\therefore y=xy$

is the required equation.