I want to find the a 3 term perturbation solution of $$(1+x)^3 = \epsilon x\tag{i}$$ where $\epsilon \ll1$.
Direct substitution of the regular perturbation series $x = x_0 + \epsilon x_1 + \epsilon^2x_2$ into (i) does not work.
I think the solution has the form: $x = x_0 + \epsilon^{1/3}\cdot x_1 + \epsilon^{2/3}\cdot x_2$. This seems to work, but I'm not sure it is correct.
TIA, Matt.