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 ?