The asymptotic solution of the three-band bisingularly problem (Q2399806)

Consider the bisingularly perturbed Dirichlet problem \[ \begin{gathered} \varepsilon^3 y''+ x^3 y'+ (x^3-\varepsilon)y= 0\quad\text{for }0<x<1,\\ y(0)= a,\quad y(1)= b,\end{gathered}\tag{\(*\)} \] where \(\varepsilon\) is a small parameter, \(a\), \(b\) are constants. It is known that \((*)\) has an unique solution. The author proves that the solution \(y(x,\varepsilon)\) of \((*)\) has the asymptotic representation \[ y(x,\varepsilon)= e^{-x}\Biggl(be+ ae^{-{x\over\varepsilon}}-be\Biggl(1-e^{-{\varepsilon\over 2x^2}}\Biggr)+ \sum^\infty_{k=1} \varepsilon^k \pi_k\Biggl({x\over\varepsilon}\Biggr)+ \sum^\infty_{k= 1} \varepsilon^{k/2} w_k\Biggl({x\over\sqrt{\varepsilon}}\Biggr)\Biggr). \] The procedures to determine the functions \(\pi_k\) and \(w_k\) are given.