Consider the problem $$ε^2x^3 + x^2 −4 = 0$$ where $0 < ε <<1$. The two regular roots of this problem are $$x = ±2−2ε^2±5ε^4$$ Find the first two non-zero terms in the asymptotic expansion for the singular root as $ε →0^+$.
Hint: It is possible for many terms in the expansion to be $0$.
To find singular, we rescale and write $x=\delta X$ with $X=O(1)$ to give $$ε^2\delta^3 X^3 + \delta^2 X^2 −4 = 0$$
Try $\varepsilon^2 \delta ^3$ ~ $\delta ^2$ $\implies$ $\delta$ ~ $1/\varepsilon^2$.
So we have $$X^3 + X^2 -4\varepsilon^4=0$$
Write $X=X_0 +\varepsilon X_1 +\varepsilon^2 X_2+O(\varepsilon ^3)$.
$O(1):X_0^3 +X_0^2=0$ so $X_0=0,-1$
$O(\varepsilon^2):X_1=0$
$O(\varepsilon^4):X_2^2-4=0$ so $X_2=2,-2$
Have I done everything correct so far? Which solutions do we ignore and why??
For the equation $$ε^2x^3 + x^2 −4 = 0$$ you properly found $$x_1=2−2ε^2+5ε^4$$ $$x_2= -2−2ε^2-5ε^4$$ and you know that the sum of the roots is $-\frac 1{\epsilon^2}$. So $$x_3=4 \epsilon^2-\frac{1}{\epsilon^2}$$ Let us check using $\epsilon=10^{-2}$. Using Cardano, the "exact" solutions are $$x_1=+1.999800050$$ $$x_2=-2.000200050$$ $$x_3=-9999.999600$$ and the above developments give exactly the same values (for ten significant figures).