Consider the equation: $\varepsilon^{3}x+x=1$.
- Look for a rescaling of this equation in the form $x = \varepsilon^n X$ with $X \in O(1)$ and explain what value $n$ should take to find two more solutions.
I've plugged in the substitution into the original equation to get:
$$ \varepsilon^{n+3}X + \varepsilon^{n}X = 1 $$
but I don't know how to explain a value for $n$ that will yield two more solutions.
What am I supposed to be looking for here?
I would be very grateful if someone is able to point me in the right direction.