Based on the theoretical knowledge of elasticity, this paper introduces elasticity (i.e. elastic transformation) as a tool to solve differential equations for the first time, and takes a class of third-order nonlinear ordinary differential equations as an example. First, elastic transformation is used to reduce the order to second-order solvable differential equation. Second, by using the elastic inverse transformation, the analytical solutions of the original equation are obtained. Finally, the steps of the elastic reduction transformation method are summarized. The examples show that some nonlinear equations can be transferred into linear equations, and this method has theoretical and applied value which is extensive and in-depth.

This paper mainly studies the problem of solving a class of first-order nonlinear non-homogeneous ordinary differential equations with variable coefficients， which can be transformed into solvable second-order linear homogeneous variable coefficient ordinary differential equations using elastic transformation. Firstly, the paper uses elastic transformation to upgrade a class of first-order nonlinear non-homogeneous variable coefficient ordinary differential equations to second-order linear homogeneous variable coefficient ordinary differential equations, and then the general solution expression of this first-order differential equation is obtained by using the general solution of the second-order linear homogeneous ordinary differential equation with variable coefficients, elastic transformation and related operation rules. Secondly, according to the solving process of first-order nonlinear non-homogeneous ordinary differential equations with variable coefficients, the steps for solving such differential equations are summarized. Finally, examples are given to prove that it is effective, simple and feasible to solve the first-order nonlinear non-homogeneous ordinary differential equation with variable coefficients by elastic transformation. The research in this paper provides a new idea for solving a class of first-order nonlinear non-homogeneous ordinary differential equations with variable coefficients.

This paper studies the definite solution problems of partial differential equation (PDE) under two kinds of elastic coefficients (exponential function type, polynomial function type) of external boundary conditions. Then the definite solution problems are solved by Laplace transformation, the method of undetermined coefficients and Gaver-Stehfest numerical inversion equation. Firstly, the definite solution problems of linear PDE are transformed into the boundary value problems of linear differential equations in Laplace Space by Laplace transformation. Secondly, the solutions in Laplace space of the boundary value problems of linear differential equations are obtained through using the method of undetermined coefficients. Finally, Obtain the real space solution of the definite solution of PDE under three kinds of elastic coefficients of external boundary conditions by using the Gaver-Stehfest numerical inversion equation. According to the above solution steps, this paper gives two different examples and obtains the corresponding curve analysis of the definite solution problems of PDE under two kinds of elastic coefficients of external boundary conditions. The introduction of two kinds of elastic coefficients of external boundary conditions not only expands the scope of the research on the definite solution of PDE, but also improves the matching degree between the theoretical model and the actual problems.

The definite solution problems of three-interval composite nonlinear partial differential equations (PDE) under different conditions is studied in the paper. Then the definite solution problems are solved by Laplace transformation - similar constructing method - Stehfest numerical inversion equation (LT-SCM-SNI method). Firstly, the definite solution problems of three-interval composite nonlinear PDE is transformed into the boundary value problems of three-interval composite linear of ordinary differential equation (ODE) by linearization and Laplace transformation (LT). Secondly, the solution in Laplace space of the boundary value problems of three-interval composite linear of ODE is obtained through using the similar constructing method (SCM). The solution in real space of the definite solution problems of three-interval composite nonlinear PDE is obtained through using the Stehfest numerical inversion equation (SNI) and linearization equation. Finally, the nonlinear spherical seepage model of three-interval composite reservoir under infinite outer boundary conditions is solved through using the LT-SCM-SNI method. The influence of various parameters on dimensionless bottom-hole transient pressure is studied. The example shows the simplicity and practicability of the method presented in the paper.