First Integrals and Exact Solutions of a Class of Nonlinear Systems

Nonlinear nonconservative systems are not integrable for a general parametric choice, and it is difficult to obtain the first integral directly. Thus, finding novel procedures to solve these systems has been an interesting topic. To accomplish this task, this paper explores a logarithm transformation to consider the first integrals and exact solutions of a class of second-order ordinary differential equations which can be physically implemented in the circuit system. Based on the transformation, the first integrals are derived and the corresponding totally integrable first-order nonlinear differential equation is obtained under certain parameter conditions. It is found that the total integrability provides a class of exact solutions. Moreover, five types of particular examples are discussed, comparison of the exact solutions and numerical results are implemented, which demonstrates the correctness and efficiency of this proposed method.