I am trying to solve the following differential equation using boundary layer method.

$$\psi ''(z) + \frac{1}{z} \psi'(z)\left(3 - \displaystyle\frac{4}{1+(\frac{z}{zc})^8}\right)+ \left(\displaystyle\frac{m^2}{1+(\frac{z}{zc})^8}\right)\psi(z)=0$$

z varies between some finite value say, $z_{uv}$ to $\infty$. where $zc$ and $m$ are some constants, and the boundary conditions are $\psi(z_{uv})=1$ , $\psi'(z_{uv})=0$and $\psi(\infty)=0$.

This differential equation has rapidly varying solutions in the neighbourhood of $z=zc$. So there exists an internal boundary layer around $zc$. Note that $z_{uv}<zc<\infty$.

I was reading, C .M.Bender and S .A.Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Per turbation Theory, Springer, 1999. In Chapter 9, they discuss the boundary layer method, however

I can 't figure out how I can rewrite the equation in the form where there is some $\epsilon << 1$ multiplying the highest derivatives. So that I can calculate the outer and inner solutions, and match at the boundary.

Any suggestions?


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