Asymptotic Matching of an logarithmic Outer solution to an exponential growing inner solution

Hi,

I'm studying an ODE with a small parameter $\epsilon$ and I'm trying to find the solution in terms of a zeroth-order term and a boundary layer. The zeroth-order term has a logarithmic behavior near $x=0$ while the boundary layer term has an exponential (special function Bi) behavior at $+\infty$. To get all the constants I need to do some sort of asymptotic matching between the two solutions but I'm a bit at a loss as to whether this is possible.

More specifically,

the solutions that span the outer solution are the Bessel functions $J_0$ and $y_0$ and the inner solutions are the integral of a sum of the Airy functions $Ai$ and $Bi$

The problem of-course pertains to the $y_0$ and the $Bi$. The only way out that I see now is to set the coefficients in front of both functions to zero due to this problem and only use the other two. But then there's a problem with the boundary conditions.

Any advice or a reference would be greatly appreciated.

Cheers,

Yossi.

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