zbMATH — the first resource for mathematics

Asymptotic behaviour of the solutions of a certain type of the third order differential equations. (English) Zbl 0685.34064
The paper is devoted to the scalar equation \[ (1)\quad (r(t)(r(t)x')')'+2a_ 1(t)x'+(a'_ 1(t)+b_ 1(\quad t))x=0,\quad t\geq t_ 0, \] where \(r(t)>0\), \(a_ 1\in C^ 1[t_ 0,\infty)\), \(b_ 1\in C[t_ 0,\infty)\). The author produces transformations which transform the equation (1), in the cases \(\int^{\infty}dt/r(t)=\infty\) and \(\int^{\infty}dt/r(t)<\infty\), into equations of the form \((2)\quad y'''+2a(t)y'+(a'(t)+b(t))y=0,\) and presents some asymptotic properties of solutions of (1) through such properties of solutions of (2). There are some technical wants in the paper. For example, it is sufficient to assume \(r\in C[t_ 0,\infty)\), \(r(t)>0\) but then \(R\not\in C^ 4[t_ 0,\infty)\) and further, we can not write \[ r(t_ 0)(r(t_ 0)x''(t_ 0)+r'(t_ 0)x'(t_ 0))\quad but\quad [r(t)(r(t)x'(t))']_{t=t_ 0} \] (see p. 216 and 217).
Reviewer: J.Ohriska
34E05 Asymptotic expansions of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
Full Text: EuDML
[1] GREGUŠ M.: On linear differential equations of higher odd order. Proc. Equadiff II. SPN Bratislava 1969, 81-88. · Zbl 0185.15904
[2] GREGUŠ M.: Third Order Linear Differential Equations. D. Reidel Publishing Co. Dordrecht, Boston, Lancaster, Tokyo 1987. · Zbl 0602.34005
[3] PHILOS, CH. G.: Oscillation and asymptotic behaviour of third order linear differential equations. Bull. of the Institute of Mathematics Academia Sinica. Vol. 11, No.: 2, 1983, 141-160. · Zbl 0523.34028
[4] ROVDER J.: Oscillation criteria for third-order linear differential equations. Mat. Čas., 25, 1975, 3, 231-244. · Zbl 0309.34028 · eudml:29549
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.