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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
##### MSC:
 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
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##### References:
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