Suppose $y=y(x)$ is infinite continuous in $\mathbb{R}$, and $y(-1)=0$, how can we obtain the analytic solution in closed form to the following nonlinear ordinary differential equation:
$$ \left((x-10)^2+y^2\right)\left(x^2+y^2y'^2+2xyy'+2yy'+2x+1\right)=\left(x^2+y^2\right)\left(x^2+y^2y'^2+2 x y y'-20y y'-20x+100\right) $$
The resulted solution should be in implicit form. Is there any general approach in solving such kind of ODE's?
Update:
What I am asking is actually: is it possible to establish ellipse equation from only one of its properties as shown in the figure below. The light rays from one fixed point $F_1$ being reflected by the curve always focus on another fixed point $F_2$ and vice versa.
Suppose we don't know the curve is ellipse, then is it possible for us to obtain ellipse formulation only from the above relationship when being given $F_1$, $F_2$ and a point $A$ or $B$ on the curve?
I think the key now becomes how to establish the nonlinear ordinary equations or nonlinear systems to solve in polarized coordinate frame or just Descartes' frame. What should I do?
I have obtained the nonlinear ODE with the analytic solution, which can be seen here: ODE and IBCs
Though it seems there is no classic approach available to solve such an ODE, the geometric meaning here can be seen as another solving approach to such nonlinear ODE problems.
Update A solution in details has been given in the same link: ODE and IBCs