A New Analytical Method for Solving van der Pol's and Certain Related Types of Non-Linear Differential Equations, Homogeneous and Non-Homogeneous

A simple analytical treatment is developed for the differential equation ${\mathrm{(d}}^2{\mathrm{u/dt}}^2{\text{)−ε(1−u}}^2\text{)×(du/dt)+u=0}$ (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 $\text{t≧0}$ , 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 $u^2{\mathrm{(t)+u}}^{\text{′2}}\text{(t)∼4,\hspace{0.17em}t≧0,\hspace{0.17em}0<ε≪1}$ . The method and results are then extended to the more general equation ${\mathrm{(d}}^2{\mathrm{u/dt}}^2\text{)−εF(u)(du/dt)+u=0}$ , in particular, for ${\text{F(u)=1−u}}^4{\text{,\hspace{0.17em}1−2au−u}}^2$ , a=constant, also to the non‐homogeneous equation ${\mathrm{(d}}^2{\mathrm{u/dt}}^2\mathrm{)-\epsilon F(u)(du/dt)+u=}\mathrm{given\; function\; of}\mathrm{\hspace{0.17em}t}$ . 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).