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Further results on common zeros of the solutions of two differential equations. (English) Zbl 1282.34092
Summary: Two problems are discussed. In the first problem, we consider one homogeneous and one non-homogeneous differential equation and study when the solutions of these differential equations can have (nearly) the same zeros. In the second problem, we consider two linear second-order differential equations and investigate when the solutions of these differential equations can take the value 0 and a non-zero value at (nearly) the same points.
We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.
In the first problem, the results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Regarding the second problem, the results also show all pairs of such equations having solutions taking the value 0 and a non-zero value at (nearly) the same points.
MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
34M03 Linear ordinary differential equations and systems in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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References:
[1] doi:10.1090/S0002-9939-1986-0831388-8 · doi:10.1090/S0002-9939-1986-0831388-8
[2] doi:10.1186/1029-242X-2011-134 · Zbl 1276.34078 · doi:10.1186/1029-242X-2011-134
[3] doi:10.4153/CMB-1974-064-0 · Zbl 0314.30021 · doi:10.4153/CMB-1974-064-0
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