I am given with the following:
$$\epsilon y'' + e^x (xy' -y ) = x^2,\quad -1<x<1,\quad 0< \epsilon \ll 1,\text{ and }y(-1)=1,\ y(1)=-1 $$
We learned that in such a case we can expect an interior layer around $x=0$ .
I need to find the leading term of the asymptotic expansion.
Outer solutions are easily found: $$y_0 ^L =(e-1)x -xe^{-x},\quad y_0 ^R = \left(\frac{1}{e} -1 \right) x -xe^{-x}.$$
The inner solution is: $$c_1 \frac{x}{\sqrt{\epsilon}}\int _{0}^{\frac{x}{\sqrt{\epsilon}}} e^{-\frac{s^2}{2} } ds - c_1 e^{- \frac{s^2}{2} } + c_2 \frac{x}{\sqrt{\epsilon}}.$$
I have no idea how to match the inner and outer solutions now.
Will someone please help me figure this out?
Thanks in advance !