PAINLEVE ANALYSIS AND INTEGRABILITY OF THE DAMPED ANHARMONIC-OSCILLATOR EQUATION

The Painleve analysis is applied to the anharmonic oscillator equation x + dx over dot + Ax + Bx(2) + Cx(3) = 0. The following three integra ble cases are identified: (i) C = 0, d(2) = 25 A/6, A > 0, B arbitrary , (ii) d(2) = 9A/2, B = 0, A > 0, C arbitrary and (iii) d(2) = -9A/4, C = 2B(2)/(9A), A < 0, C < 0, B arbitrary. The first two integrable ch oices; are already reported in the literature. For the third integrabl e case the general solution is found involving elliptic function with exponential amplitude and argument.