An Approximation to the Periodic Solution of a Differential Equation of Abel

The Abel equation, in canonical form, is y$^{'}$ = sint- y$^{3 }$(*) and corresponds to the singular ($\varepsilon $ --$>$ 0) limit of the nonlinear, forced oscillator $\varepsilon $y$^{''}$ + y$^{'}$ + y$^{3}$ = sint, $\varepsilon ->$ 0. (**) Equation (*) has the property that it has a unique periodic solution defined on (-$\infty $,$\infty )$. Further, as t increases, all solutions are attracted into the strip $\vert $y$\vert \quad <$ 1 and any two different solutions y$_{1}$(t) and y$_{2}$(t) satisfy the condition \begin{center} Lim [y$_{1}$(t) - y$_{2}$(t)] = 0, (***) \end{center} t --$> \quad \infty $ and for t negatively decreasing, each solution, except for the periodic solution, becomes unbounded.\footnote{U. Elias, American Mathematical Monthly, vol.115, (Feb. 2008), pps. 147-149.} Our purpose is to calculate an approximation to the unique periodic solution of Eq. (*) using the method of harmonic balance. We also determine an estimation for the blow-up time of the non-periodic solutions.