I'm trying to solve this differential equation : $$ \frac{dy}{dx}= \frac{-2 y^3}{(y+1)^2(y+2)^2} $$ with the boundary condition $y(x_0)=x_0$, $x>0$, and $y(x)$ being a positive function. The integration of the equation is straightforward, however after integration, one gets a transcendental equation of the form $$a y(x)+by^2(x)+c \log(y)+\frac{d}{y}+\frac{e}{y^2}+ g(x_0)= z (x-x_0)$$ where $a,b,c , d, e,z $ are constants, and $g(x_0) $ is a function of $x_0$. I tried to solve it with Lagrange inversion theorem, however due to the non triviality of the LHS, the computation of the $n$'th derivative is very complicated, is there any other way to solve it ?