Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 79 / 1 / a / Solution

Use dominant balance for algebraic roots to separate the three scales. On writing $x=ϵX$, the equation becomes $X^3-X^2+2ϵX+2{ϵ}^3=0$. Its nonzero simple limiting root is $X=1$, so one root has $x_1\sim ϵ$.

For the two remaining roots put $x={ϵ}^{2}Y$. Division by ${ϵ}^5$ gives $ϵY^3-Y^2+2Y+2ϵ=0$. The nonzero limiting root is $Y=2$, giving $x_2\sim 2{ϵ}^2$. Finally $x={ϵ}^{3}Z$ gives, after division by ${ϵ}^6$,Its simple limiting root is $Z=-1$. These three distinct balances account for all roots of the cubic, andFor the root of smallest magnitude, set $Z=-1+bϵ+O({ϵ}^2)$. The order-$ϵ$ coefficient is $-1+2b$, so $b=1/2$. HenceEach rescaled root is simple at $ϵ=0$, justifying the regular perturbation of a simple algebraic root rather than treating the original triple root as an ordinary regular perturbation.