Today I encountered this regular perturbation problem (from a habit of doing random math problems on boring classes):
$$y''(x)=-\epsilon y(x)^2-\sin (x),y(0)=0,y'(0)=1$$
A regular perturbation series using $y_0 + \epsilon y_1 + \epsilon^2 y_2$ gives:
$$ \sin (x) + \frac{1}{8} \epsilon \left(-2 x^2-\cos (2 x)+1\right) + \epsilon ^2 \left(\frac{1}{72} \left(-9 \left(4 x^2-27\right) \sin (x)-96 x-\sin (3 x)\right)-2 x \cos (x)\right) $$ Graphically (with $\epsilon = 0.1$, green-dashed is $sin(x)$, red-dashed is the perturbation solution, and blue-solid is the numerical solution) we have the figure below. The real solution (blue-solid) seems to develop a singularity around 12, as complained by Mathematica's NDSolve command, but I don't see that the regular perturbation solution would be able to capture it -- even with more terms as they are just combinations of polynomials and trigonometry functions.
So my question is if there are other, in some sense "more advanced" perturbation methods that could capture the global picture. I have Bender's book and will look into it, meanwhile I think it's a good idea to post it here as a question, as someone may point me a more precise direction.