The numerical method of ordinary differential equations is a branch of computational mathematics.

It is a numerical method for solving definite solutions of various ordinary differential equations. The existing analysis methods can only be used to solve some special types of definite solutions. In practice, the solutions of many valuable ordinary differential equations cannot be expressed by elementary functions, and their numerical solutions are often needed. The so-called numerical solution refers to the approximation of the real solution given at a series of discrete points in the solution interval.

This promotes the emergence and development of numerical methods. As the basic content of numerical analysis, the research on numerical solutions of ordinary differential equations has developed quite well in theory, and various practical algorithms have been established and computer software has been formed. Its ideas and methods for solving problems can often be used for numerical solution of partial differential equations.

This paper mainly studies the numerical solutions of the following three kinds of definite solution problems: initial value problem, two-point boundary value problem and eigenvalue problem. The numerical solution of initial value problem is widely used and is the main content of numerical solution of ordinary differential equations.

Scholars who have made outstanding contributions in this field include Dahlkvist (G.), Butscher (J.C.) and Jill (Gear, C.W.). The research on two-point boundary value problem and eigenvalue problem is weak, among which Keller's work has great influence.