Summary
The family of trajectories (in the phase plane) of van der Pol's equation is a one parameter family of curves; no explicit analytical expression for it is known. The same is true for the orthogonal trajectories of this family. The differential equation of the orthogonal curves is solved here exactly in terms of modified Bessel functions. Two curves of the orthogonal family are found to be especially simple, and a graph is given which would permit plotting the solutions of van der Pol's equation with an accuracy higher than that attainable with the usual methods.
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References
Bickart, T. A., J. Franklin Inst. 278 (1964) 256.
Urabe, M., Numerical study of periodic solutions of the van der Pol equation, International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, ed. by J. P. LaSalle and S. Lefschetz, 184–192, Academic Press, London 1963.
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Abdelkader, M.A. On van der Pol's equation. Appl. Sci. Res. 18, 112–115 (1968). https://doi.org/10.1007/BF00382341
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DOI: https://doi.org/10.1007/BF00382341