Find the Particular solution to the Cauchy problem (non-linear $1$st-order ODE variable coefficients) let $y(x)$ be a continuous function $0\le y(x) \le1$ over the support $x_\min\le x \le x_\max$, for $x_\min < x_\max$ strictly positive values. 
let $y'(x)$ be the first order derivative of $y(x)$.
let $y(x)^k$ be the $k$-th power of the function $y(x)$.
Solve the Cauchy problem:
$$a(x)y'(x) = b(x)y(x)^{-3} - cy(x)^{-2} - y(x)^{-1}$$
$$y(x_0) = y_0$$
where $a(x)=a_1 + a_2x$ with $a_1>0, a_2<0$ and $b(x)=b_1 + b_2x$ with $b_1<0, b_2>0$ 
and $x_0$, $y_0$, $c$ are positive constant real values. 
I am interested in a solution such that $y'(x)<0$. I proved existence and uniqueness, but I am looking for the analytic expression of $y(x)$. 

My academic interest are PDEs and ODEs. Personally, I'd like to say that there would be explicit solutions for such nonlinear ODE in the first place.
