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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?
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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
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