I was wondering what techniques we can use when there is no $y(x)$ in the ODE. Specifically, we've covered singular and regular perturbations for boundary layers but there's no example with a missing y(x).
For example:
Find the lowest-order uniform approximation to the boundary-value problem \begin{align}\label{bvp} \begin{cases} \epsilon y''(x) + x y'(x) = x \cos(x),\\ y(1) = 2,\\ y(-1) = 2. \end{cases} \end{align}
For the outer solution you get $$ y_o(x)=C+\sin x $$ This can not satisfy both boundary conditions simultaneously, so you get boundary or fast layers.
These are possible at both the boundaries $x=\pm 1$ or at the single exceptional inner point $x=0$.
At $x=-1$ one would set $x=-1+\delta X$, $Y(X)=y(x)$ to get $\delta=ϵ$ and $$ Y''(X)-Y'(X)=0, ~~ Y(0)=2, Y(\infty)\text{ bounded } $$ which only has the trivial constant solution. The situation at $x=+1$ is a mirror image of this.
At $x=δX$ the balance and equation reduce to $δ^2=ϵ$, $$ Y''(X)+XY'(X)=0\implies Y(X)=C+DS(X) $$ with $S$ a sigmoid function that can be parametrized to look similar to the hyperbolic tangent, $S'(X)=k\exp(-X^2/2)$, $S(\pm\infty)=\pm 1$.
In total, after applying the constant balancing formalism of your choice, the combined first-order approximation should look like $$ y(x)=2+\sin(x)-\sin(1)S(ϵ^{-1/2}x). $$