Asymptotic theory for third-order differential equations. (English) Zbl 0697.34050
For the differential equation \((q(qy')')'=(py')'+ry\) it is shown that under certain assumptions there exist three solutions with the asymptotic behaviour
\[
y_ k\sim (pr)^{-1/4}\exp \int^{x}_{a}\lambda_ k(t)dt\quad (k=1,2),
\]
\[ y_ 3=o\{(p/rs)^{1/2} \exp \int^{x}_{a}\lambda_ 3(t)dt\}\quad for\quad x\to +\infty, \] where \(\lambda_ k\) \((k=1,2,3)\) are the solutions of \(q^ 2\lambda^ 3=p\lambda^ 2+r\) with the asymptotic representations \(\lambda_ 1\sim i(r/p)^{1/2}\), \(\lambda_ 2\sim -\lambda_ 1\), \(\lambda_ 3\sim p/q^ 2\).
\[ y_ 3=o\{(p/rs)^{1/2} \exp \int^{x}_{a}\lambda_ 3(t)dt\}\quad for\quad x\to +\infty, \] where \(\lambda_ k\) \((k=1,2,3)\) are the solutions of \(q^ 2\lambda^ 3=p\lambda^ 2+r\) with the asymptotic representations \(\lambda_ 1\sim i(r/p)^{1/2}\), \(\lambda_ 2\sim -\lambda_ 1\), \(\lambda_ 3\sim p/q^ 2\).
Reviewer: L.Berg
MSC:
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
| 34E99 | Asymptotic theory for ordinary differential equations |
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