Characteristic differential equation is a mathematical term published in 1993.

The characteristic equation of differential equation is y'+p (x) y'+q (x) y = f (x). Characteristic equations are some equations introduced for studying corresponding mathematical objects, which are different for different mathematical objects, including series characteristic equations, matrix characteristic equations, differential equation characteristic equations, integral equation characteristic equations and so on.

Differential equation refers to the relationship with unknown function and its derivative. Solving differential equations means finding unknown functions. Differential equations are developed by calculus. Differential equations are widely used and can solve many problems related to derivatives. Many kinematics and dynamics problems involving variable forces in physics, such as falling bodies with air resistance as speed function, can be solved by differential equations.

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Characteristic equations are some equations introduced for studying corresponding mathematical objects, which vary with different mathematical objects, including series characteristic equations, matrix characteristic equations, differential equation characteristic equations, integral equation characteristic equations and so on.

Characteristic equations are some equations introduced for studying corresponding mathematical objects, which vary with different mathematical objects, including series characteristic equations, matrix characteristic equations, differential equation characteristic equations, integral equation characteristic equations and so on.

In the research and discussion of delay differential equations, it is the most classical and important problem to analyze the stability and bifurcation of the equations, which requires discussing the roots of the characteristic equations of differential equations. This paper summarizes several common dividing methods of differential equation characteristic equation, and compares and analyzes the advantages and disadvantages of several dividing methods through examples, which is of great significance for further discussion and application of several dividing methods.

As we all know, the general Riccati equation and the second-order linear equation with variable coefficients are not integrable. In this paper, reference [3] is improved and expanded, and some new integrable types of Riccati equation and second-order equation are given. Many famous integrable types and integrable results in classical and modern times are summarized by unified equations or theorems, and the concepts of resolvent function, characteristic constant, characteristic equation and discriminant are introduced to make the solutions of these equations "universal".