Structure of the solution set of some first order differential equations of comparison type

Hallam, T. G.; Heidel, J. W.

doi:10.1090/S0002-9947-1971-0281995-2

Structure of the solution set of some first order differential equations of comparison type
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by T. G. Hallam and J. W. Heidel PDF

Trans. Amer. Math. Soc. 160 (1971), 501-512 Request permission

Abstract:

The comparison principle is a powerful tool that has a wide variety of applications in ordinary differential equations. The results of this article describe the geometric structure of the solution space of some first order scalar differential equations that may arise in the comparison method. A quite general class of differential equations is found to have a similar solution set configuration as the differential equation of separable variable type. One of the main results establishes, under certain conditions, that there is a unique unbounded solution of the first order differential equation which exists on an interval of the form [${t_0},\infty$). Furthermore, this unbounded solution separates the solutions that are bounded on [${t_0},\infty$) from those that are not continuable to all $t > {t_0}$.

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