Consider the equation;
determine the two-term outer, inner and uniform expansions assuming that $0<\epsilon<<1$ $$\epsilon \frac{d^2y}{dx^2}+\frac{dy}{dx}+y=0, \hspace{5mm} y(0)=0,\hspace{2mm}y(1)=1$$
I have worked out the exact solution of why but don't know if that applies here? I have also found by setting $\epsilon=0$ that the first order outer layer term is $\exp(1-x)$. I know how to work out the first term for the inner layer aswell, but could someone explain how to work out the second terms.
To get the first term, you took the smallest powers of $\epsilon$. To get the second terms, take the second smallest terms. I'll go through the outer solution.
Let $y=y_0+\epsilon y_1+O(\epsilon^2)$, so $$ \epsilon y_0''+\epsilon^2 y_1'+y_0'+\epsilon y_1'+y_0+\epsilon y_1=0. $$ The smallest power of $\epsilon$ is $0$, so the leading order equation is $$y_0'+y_0=0,$$ the next order equation is $O(\epsilon)$, $$y_0''+y_1'+y_1=0,$$ and you can continue this as far as you want.
Once you know $y_0$, you can use it to find $y_1$, then use that to find $y_2$ and so on.