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Zbl 0964.34043
Olver, F.W.J.
On the uniqueness of asymptotic solutions to linear differential equations. (English)
[J] Methods Appl. Anal. 6, No.2, 165-174 (1999). ISSN 1073-2772

The author considers the $n$th-order $(n\ge 2)$ linear homogeneous differential equation $$w^{(n)}+ f_{n- 1}(z) w^{(n- 1)}+\cdots+ f_0(z) w=0\tag 1$$ in the neighborhood of infinity, where $f_\ell(z)$, $\ell= 0,1,\dots,n- 1$, are analytic at infinity. Formal solutions to (1) are given by $$e^{\lambda_j z} z^{\mu_j} \sum^\infty_{s= 0} a_{sj}/z^s,\quad j= 1,2,\dots, n,$$ with $a_{0j}= 1$. The author restricts his attention to the case in which the $\lambda_j$ are distinct. From the $\lambda_j$'s the author classifies explicit solutions and implicit solutions. Here, asymptotic properties of both solutions are discussed. One of the main results is that explicit solutions can be defined uniquely by their asymptotic behavior along a single ray.
[Katsuya Ishizaki (Saitama)]

MSC 2000:: *34E05 Asymptotic expansions (ODE)
34M25 Formal solutions, transform techniques

Keywords: asymptotic solutions; linear homogeneous differential equation; explicit solutions; implicit solutions; asymptotic properties

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