I am having difficulty with a two-variable singular perturbation analysis on a set of ODE's. The key difficulty is also present in the following, embarrassingly simple problem:
If $x\sim \mathcal{O}(1)$ and satisfies
$$ 0 = \varepsilon \delta x^2 + \varepsilon x + \delta (x-1) $$
where $\varepsilon \ll \delta \ll 1$, show that $x$ satisfies
$$ \varepsilon x + \delta (x-1) + \mathcal{O}(\varepsilon\delta)=0 $$
or, rearranging,
$$ x = \frac{\delta}{\varepsilon +\delta} + \mathcal{O}(\varepsilon) $$
To me, it is 'obvious' that this holds for at least one of the solutions to the quadratic equation, however I don't know how to choose a perturbation for small $\varepsilon$ and $\delta$ that gives exactly this answer! I need this expression in particular, as in the ODE case it leads to a particularly simple form which may be solved analytically.
If I begin with a perturbation of the form: $$ x = x_0 + \varepsilon x_{\varepsilon} + \delta x_{\delta} + \varepsilon^2 x_{\varepsilon\varepsilon} + \delta^2 x_{\delta\delta} + \varepsilon\delta x_{\varepsilon\delta} + \mathcal{O}(\delta^3) $$ then I would hope that the highest order approximation would give: $$ \varepsilon x_0 + \delta (x_0 - 1) + \mathcal{O}(\delta^2) = 0 $$ which is the same as the approximate expression derived intuitively above. However, because $\varepsilon \ll \delta$, I cannot exclude the possibility that $\varepsilon \sim \delta^2$, in which case I cannot justify excluding the $\delta^2$ term. Hence we have:
$$ \varepsilon x_0 + \delta (x_0 - 1) + \delta^2x_{\delta} + \mathcal{O}(\varepsilon\delta) = 0 $$
Rearranging gives $$ x_0 = \frac{\delta}{\varepsilon+\delta} - \frac{\delta^2x_{\delta}}{\varepsilon + \delta} + \mathcal{O}(\varepsilon) $$ and we have an annoying unknown, $x_{\delta}$, in an expression I want to just contain $x_0$! The first-order expression similarly contains $x_{\delta\delta}$ terms, and on it goes.
My question is, is there a way to derive the expression
$$ x = \frac{\delta}{\varepsilon +\delta} + \mathcal{O}(\varepsilon) $$
via a perturbation expansion, which would then allow for an explicit expression for the error term to order $\mathcal{O}(\varepsilon^2)$?
Thanks.
