Wh. Steeb et N. Euler, EXTERNALLY DRIVEN NONLINEAR OSCILLATOR, PAINLEVE TEST, 1ST INTEGRALS AND LIE SYMMETRIES, Zeitschrift fur Naturforschung. A, A journal of physical sciences, 48(8-9), 1993, pp. 943-944
For arbitrary constants c1, c2 and an arbitrary smooth functions f the
driven anharmonic oscillator d2u/dt2 + c1 du/dt + c2u + u3 = f(t) can
not be solved in closed form. We apply the Painleve test to obtain the
constraint on the constants c1, c2 and the function f for which the e
quation passes the test. We also give the Lie symmetry vector field an
d first integrals for this equation.