Abstract: There is a great research space for applying differential equation model to solve practical problems. This paper mainly introduces the principle of differential equation, the enlightenment of differential equation thought to solving practical problems, and the cases of using differential equation model to solve specific problems in real life, in order to put forward some thoughts after learning differential equation theory.

Keywords: differential equation; Model; App application

For the changes in the real world, people often pay attention to the rate of change between variables, or the development law of speed, acceleration and position with time, which can generally be written as (partial) differential equations or equations.

Therefore, in practical problems, many problems can be modeled by differential equations, involving physics, chemistry, astronomy, biology, mechanics, politics, economy, military, population, resources and other fields.

1. mathematical principle analysis of differential equation

In elementary mathematics, there are many kinds of equations, such as linear equation, exponential equation, logarithmic equation, triangular equation and so on. But they can't solve all the practical problems.

To study practical problems, we must seek one or several unknown equations that satisfy certain conditions.

The basic idea of this kind of problem has many similarities with the idea of solving equations in elementary mathematics, but there are still many differences in the form of equations, the specific methods of solving them and the nature of solving them. In order to solve this kind of problems, differential equations came into being.

Differential equation is a basic course for many science and engineering majors. Differential equations and calculus are produced simultaneously. From the beginning, it has become a powerful tool for human beings to understand and transform the world. With the development of production practice and science and technology, this subject has developed into an important branch of mathematical theory.

With the increasingly active activities of mathematical modeling, using differential equations to establish mathematical models has become an indispensable method and tool to solve practical problems.

Mathematical model is to make some necessary assumptions about specific objects and specific purposes in the real world, and use appropriate mathematical tools to get a mathematical structure. Simply put, it is a mathematical expression of the essence of a certain feature of the system (or a description of some real world in mathematical terms), that is, using mathematical formulas (such as functions, graphs, algebraic equations, differential equations, integral equations, difference equations, etc.). ).

Second, the method and process of applying differential equation model to practical problems are summarized.

When studying practical problems, it is often related to the rate of change or derivative of some variables, so the relationship between variables is the differential square model.

The differential equation model reflects the indirect relationship between variables, so to get the direct relationship, we must find the differential equation.

There are usually three methods or forms for solving differential equations, namely analytical solution, numerical solution (approximate solution) and qualitative theoretical method.

There are usually three methods to establish a differential equation model. One is to establish a differential equation model by using theorems in mathematics, mechanics, physics, chemistry and other disciplines or laws tested by experiments. The second is to find out the relationship between infinitesimals by using known theorems and laws. Different from the first method, this law is directly applied to infinitesimal instead of function and its derivative.

Thirdly, in the practical problems of biology, economics and other disciplines, the regularity of many phenomena is not very clear, even if understood, it is extremely complicated. When modeling, the actual phenomenon is simulated under different assumptions, and the differential equation which can approximately reflect the problem is established. Then, the established equation and its solution are solved or analyzed mathematically, and then compared with the actual situation to test whether this model can describe and simulate some actual phenomena.

In the process of establishing mathematical differential equations, the first step is usually to analyze specific practical problems, find out the relationship between variables, then make model assumptions, replace the elements of practical problems with mathematical concepts, then set symbols and simplify calculations, so as to establish models and solve them, and finally verify the previous problem analysis and model assumptions with the results of solution, and then apply and evaluate the models after verification.

Thirdly, the application fields of differential equation model are summarized, and specific cases are analyzed.

In terms of application fields, the application fields of general differential equations are mainly divided into five aspects: society and market economy, differential model analysis of war, population and animal world, infection and diagnosis of diseases, and natural science. If detailed, social and market economy includes differential equation model of comprehensive national strength, differential equation model of inducing investment and accelerating development, differential equation model of economic adjustment, differential equation model of advertising and differential equation model of price.

The differential model of war includes the differential equation model of arms race, the differential equation model of war, the differential equation model of operational survival possibility and the model of war prediction and evaluation. The field of population and animal world includes single species model and developing single species model, jungle law model, competitive exclusion model of two species in the same niche, unregulated fish fishing model, population prediction and control model.

The field of disease transmission and diagnosis includes the differential equation model of AIDS epidemic, the differential equation model of diabetes diagnosis, the differential equation model of iodine in human body, and the distribution and elimination model of drugs in the body; The natural science field includes the differential equation model of satellite motion, the differential equation model of space roll control, the differential equation model of nonlinear vibration, the differential equation model of self-excited oscillation of PLC circuit and the differential equation model of tracking and chasing problem.

Although it is complicated to list and summarize the application fields of the above differential equations, in fact, all the processes of differential equation modeling are strictly in accordance with the processes of problem analysis, model hypothesis, symbol setting, model establishment, model solution and model verification. Let's analyze it in detail with a case:

Such as the jungle law differential equation model.

There is a cruel competition for survival among all kinds of creatures living in the same environment.

Imagine an island where foxes and rabbits live. Foxes eat rabbits, and rabbits eat grass. The grass is so rich that rabbits don't worry about eating it, so they breed in large quantities. The more rabbits there are, the more foxes are delicious, and the more foxes there are. However, with the increase in the number of foxes, a large number of rabbits were eaten, and the total number of foxes went hungry again, which reduced the total number of rabbits. At this time, rabbits are relatively safe, so the total number of rabbits increases.

In this way, the number of foxes and rabbits alternately increases and decreases, and the cycle is endless, forming a dynamic balance of ecology.

Then, how to describe and predict the next stage by establishing a mathematical model? On this issue, there is a variable relationship between the number of rabbits and the number of foxes at a certain moment:

Among them, ax means that the reproduction speed of rabbits is directly proportional to the number of existing rabbits, and -bxy means the speed at which foxes meet rabbits and rabbits are eaten; -cy means that the death rate of foxes caused by the competition for food among the same species is directly proportional to the total number of foxes; Dxy represents the speed at which foxes and rabbits meet, which is good for foxes and increases reproduction.

Four. conclusion

The application of differential equation model solves many problems that are difficult to calculate in reality. It is the development trend of applying mathematics to life to scientifically model the control of the development law of things. As the majority of students studying mathematics at school, mastering basic professional skills is an important way to obtain employment in the future and realize their own value.

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