ABSTRACT In order to solve a class of third‐order nonlinear ordinary differential equations with variable coefficients, this paper first reduces the equation to the second‐order Laguerre equation by elastic transformation. Secondly, the similar structure of the solution of the Laguerre equation is derived by using a similar construction method. Finally, the inverse elastic transformation is applied to the similar structure of the solution of the Laguerre equation to obtain the similar structure of the solution of the original third‐order nonlinear ordinary differential equation. The application of the elastic transformation method broadens the solvable classes of ordinary differential equations with variable coefficients, and the similar structure method simplifies the solving steps. The combination of these two methods provides a new approach to solving the problems of nonlinear ordinary differential equations with variable coefficients.
Zheng et al. (Sun,) studied this question.