Asymptotical properties of the Wronski determinant of a certain class of linear differential equations of the 2nd order. (English) Zbl 0594.34055

Hylán, Jaroslav

Čas. Pěst. Mat. 110, 13-18 (1985).

The author studies the differential equation \(y''+[\ell^ r- q(x,\ell)]y=0\) and its solutions \(\phi\) (x,\(\ell)\), \(\psi\) (x,\(\ell)\) satisfying initial conditions \[ \phi (0,\ell)=\alpha_ 1,\quad \phi '(0,\ell)=\alpha_ 2,\quad \alpha^ 2_ 1+\alpha^ 2_ 2>0, \]
\[ \psi (a,\ell)=\beta_ 1,\quad \psi '(a,\ell)=\beta_ 2,\quad \beta^ 2_ 1+\beta^ 2_ 2>0, \] where \(q(x,\ell)=\sum^{r}_{\nu =0}a_{\nu}(x)\ell^{\nu}\), \(a_{\nu}(x)\) are real functions continuous on the interval [0,a], r is a natural number, \(r'<r/2\) an integer and \(\ell\) a complex variable. The asymptotical behavior of the Wronski determinant of \(\phi\) (x,\(\ell)\) and \(\psi\) (x,\(\ell)\) for \(| s| =\rho \to +\infty\) is investigated, where \(s=\ell^{r/2}\).

Reviewer: J.Kato

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

second order differential equation; asymptotical behavior of the Wronski determinant