Abstract
A simple analytical treatment is developed for the differential equation (d2u/dt2)−ε(1−u2)×(du/dt)+u=0 (van der Pol), in order to study the behavior of its solution assumed bounded in (0, ∞). It is shown, without any further assumption, that, if ε≪1, u(t) can be approximated closely, for t≧0, by a simple oscillation with amplitude 2. This is a more precise form of a statement due to van der Pol. It is further shown that u2(t)+u′2(t)∼4, t≧0, 0<ε≪1. The method and results are then extended to the more general equation (d2u/dt2)−εF(u)(du/dt)+u=0, in particular, for F(u)=1−u4, 1−2au−u2, a=constant, also to the non-homogeneous equation (d2u/dt2)−εF(u)(du/dt)+u=given function of t. The analytical results obtained in this paper show a remarkable agreement with those obtained for the same equations by mechanical means (on the differential analyzer).
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