I am trying to solve a textbook problem frim Hinch: Perturbation methods: $$ \varepsilon^2 y'''' - y'' + 1 =0\\ y = y' = 0 \mbox{ for } x = 0 \mbox{ and }1. $$ ] I need to find two terms in $\varepsilon$ in outer region and match it to inner solution near both boundaries.
I tried to find solution in outer region in the form of $$ y_{outer} = y_0 + \varepsilon y_1 + \mathcal O(\varepsilon^2), $$ which yielded: $$ y_{outer} = \left[\frac{x^2}{2} + a_1 x + a_2 \right] + \varepsilon \left[a_3x + x_4 \right]. $$
Next I found inner solution in re-scalled variabe $X=x/\varepsilon$ near $x=0$, which was: \begin{align} Y(X) =& B_1 \left(e^X - X - 1 \right) + B_2\left(e^{-X} + X -1 \right)\\ &+ \varepsilon \left[ C_1 \left(e^X - X - 1 \right) + C_2\left(e^{-X} + X -1 \right)\right] \end{align}
However, I am unable to match these two expansions. What am I missing?