$y''+2\epsilon y(y+1)=x$, where $y(0)=0$, $y'(0)=0$

Question

I'm trying to solve $y''+2\epsilon y(y+1)=x$, where $y(0)=0$, $y'(0)=0$ using perturbation theory.

Using the substitution $y=y_{0}+y_{1}\epsilon$ I got the series $y=\frac{1}{6}x^{3}-\epsilon(\frac{1}{36}x^{5}+\frac{1}{1008}x^{8})+O(\epsilon ^{2})$.

I used a spreadsheet to solve the ODE with Euler's method and I'm not convinced that this perturbation series matches the numerical solution.