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Perturbation solutions of the Carson–Cambi equation

✍ Scribed by Barbara Epstein; Richard Barakat

Publisher
Elsevier Science
Year
1977
Tongue
English
Weight
586 KB
Volume
303
Category
Article
ISSN
0016-0032

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✦ Synopsis

The periodic differential equation (1 + E cos t)y"+ py = 0, hereby termed the

Carson-Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided (e( < 1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p-e plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to e3. Two-timing theory is employed to obtain solutions for values of k near the critical points

k = *f, *$, *$. Three-timing is employed to extend the solution near k = *f. The solutions of the Carson-Cambi equation are compared with the solutions of the corresponding Mathieu equation.

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