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Exact solutions and integrability of the Duffing-van der Pol equation. (English) Zbl 1425.34105
This paper studies the integrability of the Duffing-van der Pol equation \[ X_{tt} + (\alpha + \beta X^2) X_t - g X + X^3 = 0.\tag{1} \] The author tries to find the exact solution of this equation by a series of transformations.
However, I cannot find the conclusion of this paper, and there are many mistakes in the calculation and notations in the paper.
For example, from (2.5)–(2.12) (equation number in the paper), the author introduces a transformation \(X(t) = \sqrt{V(t)}\), which gives \[ V V_{tt} - \frac{1}{2} V_t^2 + \alpha V V_t + \beta V^2 V_t -2 g V^2 + 2 V^3 = 0.\tag{2} \]
In (2.20), the author claims a solution in form \[ V(t) = \frac{g}{1 + \exp(-\frac{3}{2} \beta g(t-t_0))}. \] However, this is NOT a solution of (2)!

MSC:
34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
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