On the asymptotic behavior of solutions of a certain second order ordinary differential equation (Q1084573)

A fundamental pair of solutions is given for the differential equation \(u_{tt}+k^ 2(t^ 2+y^ 2)u+kc=0,\) where u(t,y) is the unknown function, \(k>0\), c is a complex constant and (t,y) is in \(R^ 2\). The novelty lies in its derivation and the corresponding asymptotic expansion, which is different than the given usually. Let \(\epsilon\) and \(\delta\) be the symbols \(+\) or -. Then, the fundamental theorem states that for each \(\epsilon \in \{+,-\}\) there is a pair of fundamental solutions \(v^{\delta}_{\epsilon}(t,y,k)\), defined and infinitely differentiable in some domain \(D_{\epsilon}\) of the (t,y,k) space, and their asymptotic expansions are given (they are related to confluent hypergeometric functions). The given results are expected to generalize to solutions of the partial differential equation \(u_{tt}- u_{xx}+(t^ 2+x^ 2+y^ 2)k^ 2+kc=0.\)