I am working on some firm optimization problem and got to the following differential equation.
$\frac{dy}{dx} = \frac{(y-x)(1-y)}{(c-x)(1-2y+x)}$ with $x,y \in [0, 1]$ and c is a constant with $0<c\leq1$.
I know the obvious solution $y=1$ but I was wondering if there is any other solutions. Also I am looking for a generic solution that depends on any $c$ within the specified interval.
Thanks.
$$\frac{dy}{dx} = \frac{(y-x)(1-y)}{(c-x)(1-2y+x)}$$ Let $u(x)=1-2y+x\qquad;\qquad y=\frac12(1+x-u)$ $$\frac12(1-u')=\frac{(1-x-u)(1-x+u)}{4(c-x)u}$$ $$uu'=1-\frac{(1-x)^2-u^2}{2(c-x)}$$ Let $u(x)=\frac{1}{v(x)}$ $$v'=\frac{(1-x)^2-2(c-x)}{2(c-x)}v^3-\frac{1}{2(c-x)}v$$ This is an Abel's differential equation of the first kind : http://mathworld.wolfram.com/AbelsDifferentialEquation.html .
Don't confuse with Abel's differential equation identity.
To go further, see : https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf $$v(x)=\pm\sqrt{\frac{3(c-x)}{C_1+x(6c-x^2-3)}}$$
To be checked. This is an arduous calculus. Not sure about correctness.