# Find leading-order asymptotic approximatons to the solution of epsilon y''+Cosh[x]y'- y == 0

I am trying to solve this boundary value problem and obtain the leading-order approximations using asymptotic matching. But I got my solution wrong and I am stuck along the way, struggling to find any mistakes that I have made. I would really appreciate any help!

Find leading-order asymptotic approximations to the solution of

epsilon y'' + Cosh[x] y' - y == 0, y[0]==y[1]==1.


Compare with the numerical solution for epsilon = 0.05.

This is my work:

eq_out = Cosh[x] y'[x] - y[x] == 0;

sol_out = DSolve[{eq_out, y_out[1]==1}, y_out, x]
(*
{{y_out->Function[{x}, e^(1-x)]}}
*)


Then I solve for the inner equation:

Clear[x];
eq_in = Expand[epsilon y''[x]+Cosh[x] y'[x]-y[x]==0 /.
{y''[x]->1/epsilon^2 Y''[w], y'[x]->1/epsilon Y'[w],y[x]->Y[w]}]


Now I take the limit epsilon -> 0

eq_in = Y[w]==0;
sol_in = DSolve[{eq_in, Y[0]==0}, Y,w][[1]]
(*
{Y->Function[{w},0]}
*)


My questions are:

1. What is wrong with my work?
2. If outer solution is 0, then how can I go about finding the overlapping region?
3. How can I get the total solution?
-
You should not use symbol names like sol_in and sol_out, the _ character has built-in meaning in Mathematica. –  Silvia Jun 14 '13 at 5:10