On the uniqueness of asymptotic solutions to linear differential equations.

*(English)*Zbl 0964.34043The author considers the $n$th-order $(n\ge 2)$ linear homogeneous differential equation

$${w}^{\left(n\right)}+{f}_{n-1}\left(z\right){w}^{(n-1)}+\cdots +{f}_{0}\left(z\right)w=0\phantom{\rule{2.em}{0ex}}\left(1\right)$$

in the neighborhood of infinity, where ${f}_{\ell}\left(z\right)$, $\ell =0,1,\cdots ,n-1$, are analytic at infinity. Formal solutions to (1) are given by

$${e}^{{\lambda}_{j}z}{z}^{{\mu}_{j}}\sum _{s=0}^{\infty}{a}_{sj}/{z}^{s},\phantom{\rule{1.em}{0ex}}j=1,2,\cdots ,n,$$

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.

Reviewer: Katsuya Ishizaki (Saitama)

##### MSC:

34E05 | Asymptotic expansions (ODE) |

34M25 | Formal solutions, transform techniques (ODE in the complex domain) |