Asymptotic solution of a singularly perturbed Cauchy problem with a turning point (Q2663212)

Using the generalized method of boundary-layer functions, the authors construct a uniform asymptotic expansion of the unique solution of the Cauchy problem for a singularly perturbed problem with a turning point, \[ \varepsilon y''_{\varepsilon}(x)+x^n y_{\varepsilon}(x)=f_{\varepsilon}(x),\ 0 < x < T, \] \[ y_{\varepsilon}(0)=a,\ y'_{\varepsilon}(0)=b, \] where \(\varepsilon>0\) is a small parmeter, \(n\) is a fixed natural number, \(T,a,\) and \(b\) are constants, and \[ f_{\varepsilon}(x)=\sum\limits_{k=0}^{\infty}\varepsilon^k f_k(x), \ f_k\in C^{\infty}[0,T], \ f_0(0)\neq0. \]