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On the complete integrability and linearization of nonlinear ordinary differential equations - Part II: Third order equations
We introduce a method for finding general solutions of third-order nonlinear
differential equations by extending the modified Prelle-Singer method. We
describe a procedure to deduce all the integrals of motion associated with the
given equation so that the general solution follows straightforwardly from
these integrals. The method is illustrated with several examples. Further, we
propose a powerful method of identifying linearizing transformations. The
proposed method not only unifies all the known linearizing transformations
systematically but also introduces a new and generalized linearizing
transformation (GLT). In addition to the above, we provide an algorithm to
invert the nonlocal linearizing transformation. Through this procedure the
general solution for the original nonlinear equation can be obtained from the
solution of the linear ordinary differential equation.Comment: Submitted to Proceedings of the Royal Society London Series A, 21
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