Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations

Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.