The asymptotic expression of the solution of the Cauchy's problem for a higher order linear ordinary differential equation when the limit equation has singularity (Q1333828)

This paper extends previous results of the authors concerning initial value problems involving a small parameter. The previous work applied to equations of the order 2 or 4, and the present work applies to equations of arbitrary order. The equation is of the form \(\varepsilon^{n + 2} y^{(n + 1)} + a_ 0 (x)y^{(n)} + \cdots + a_ n (x)y = f(x)\), and when \(\varepsilon = 0\) the leading coefficient \(a_ 0 (x)\) has a singularity at \(x = 0\), i.e. \(a_ 0 (0) = 0\). The authors obtain asymptotic expansions for the initial value problem \[ y_ 1 (t, \varepsilon) = \sum^ n_{i = 0} \varepsilon^{{n + 2 \over n + 1}i} y_ i(t) + R_{1,n}, \quad 0 \leq x \leq \varepsilon, \] \[ y_ 2 (z, \varepsilon) = \sum^ n_{i = 0} \varepsilon^ iy_ i (z) + \varepsilon^ n \sum^{2n}_{i = 0} \varepsilon^ i v_ i(t) + R_{2n}, \quad \varepsilon \leq x \leq N. \] The remainder terms satisfy \(R_{1n} = O (\varepsilon^{n + 2})\), \(R_{2n} = O (\varepsilon^{n + 1})\), and the \(y_ i\) are determined by \[ y_ i (t) + O \bigl( P(t^{(n + 2)i}) \bigr) \exp \left[ - {t^{n + 1} \over n + 1} \right] + O \bigl( R(t^ i) \bigr), \] and \(P(t^ i)\) is a polynomial in \(t\), of order \(\leq i\). The calculation of the asymptotic expansion in the third region is analogous to a previous result of the author.