EQUIVALENT LAGRANGIANS AND THE SOLUTION OF SOME CLASSES OF NONLINEAR EQUATIONS Q+P(T)Q + R(T)Q=MU-Q2Q-1 + F(T)QN

The second-order ordinary differential equation q + p(t)q + r(t)q = muq2q-1 +f(t)q(n), where mu not-equal 1 is linearizable (sl(3, R) algebra) via a point transformation if and only if n = mu or n = 1. We construct a quadratic Lagrangian L = 1/2 A(t, q)q2 + B(t, q), which determines the point transformation Q = F (t, q) and T = G(t, q) that maps the Lagrangian to the simple completely integrable Lagrangian LBAR = 1/2 Q'2. For n = 4mu - 3 the equation admits the sl(2, R) algebra. In this case we again construct a quadratic Lagrangian and then obtain the corresponding point transformation that reduces the original Lagrangian to the representative Lagrangian LBAR = Q'/T + 1/(2TQ'). For both cases, sl(2, R) and sl(3, R), we obtain complete solutions (cf. [1, 2]).