Summary
In several fields of engineering research, particularly in the study of vibrations, electrical circuits and in some problems of fluid mechanics, approximations which lead to linear differential equations are proving inadequate. This circumstance is focussing the attention of research workers and engineers on non-linear problems. This article gives an account, without proofs, but with literature references, of methods for the qualitative integration of non-linear ordinary differential equations of the first order, i.e. for the determination of the pattern of the integral curves of such equations. The use of such geometrical methods becomes necessary in cases when the equation cannot be integrated in closed form. Simple and complex patterns associated with singular points are discussed, and criteria for their classification are given. A method of determining the asymptotic behaviour of the family of solutions is given, and criteria for the existence of closed curves in the family of solutions, as well as the occurrence of limit cycles, are discussed. A brief discussion of the Kronecker index and of the mutual relation between several singular points is added. The text is illustrated with several examples selected from the fields of vibration, compressible fluid flow and electrical circuits.
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Kestin, J., Zaremba, S.K. Geometrical methods in the analysis of ordinary differential equations. Appl. Sci. Res. 3, 149–189 (1954). https://doi.org/10.1007/BF02123900
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DOI: https://doi.org/10.1007/BF02123900