I want to find the a 3 term perturbation soln of
(i) $(1+x)^3 = ex$ where $e\ll1$


Direct substitution of the regular perturbation series $x = x\_0 + ex\_1 + e^2x\_2$ 
into (i) does not work


I think soln has the form: $x = x\_0 + e^{1/3}\*x\_1 + e^{2/3}\*x\_2$
Seems to work, but not sure it is correct


TIA,
Matt



Regular Perturbation Series soln to eqn
Since $e$ is small, the solution $x$ is close to $-1$. So write $x=-1+u$ and write your equation as $u(1-u)^{-1/3}=-e^{1/3}$. Then use the [Lagrange inversion formula](http://en.wikipedia.org/wiki/Formal_power_series#The_Lagrange_inversion_formula).


