I'm looking to find the first three terms in the perturbation expansion for all three roots of the equation
$x^3+x^2-\epsilon=0$, where $0<\epsilon \ll 1 $.
So far, by setting $x=x_0+\epsilon^\alpha x_1+ \epsilon^{2\alpha} x_2$ and equating terms of equal order, I've been able to find one root - $x_0=-1, x_1=-1,x_2=2$ and this is with $\alpha =1 $.
Could anyone help find the other two. I've tired equating higher order terms with no success. Also, when I let $y=\frac{x}{\delta}$ and get $\delta=\epsilon^{\frac{1}{3}}$, I'm still not getting anywhere after.
Thanks.