This paper explores several differential equation models in the economic system and analyses the solution and stability of the differential equation models in order to better reflect the theoretical results in mathematics into reality. From a mathematical point of view, the analysis illustrates the important role of differential equation models in economic systems.
- differential equation
- general solution
Differential equation models are extremely effective mathematical methods for solving many practical problems. As an important branch of mathematics, differential equations have been developed over many years, and their solutions and qualitative theories have become more and more perfect. They can be used to obtain solutions (or numerical solutions) of differential equations and provide enough methods to make differential equation modelling very effective and very rich in internal functions . In macroeconomics, people consider the changes of various economic parameters at a finite point in time or between infinite points in time, such as changes over time, how output, consumption levels, wage levels and capital stock levels change, to describe their own changes must resort to differential equations or difference equations.
Sales promotion: It refers to a professional activity of industrial and commercial enterprises in the economic field to tap potential customers and promote the sale of goods. It refers to the process of a series of promotional methods and activities adopted by industrial and commercial enterprises in a certain business environment for their sales targets. Supply volume: It refers to the volume of commodities that an enterprise is willing to sell and available for sale per unit time under certain price conditions. It is recorded as S. Demand: It refers to the amount of goods that consumers want to buy and have the ability to pay per unit time under certain price conditions  and is denoted as D. Equilibrium price: It refers to the price when supply and demand in the market are equal. It is recorded as
There is no periodic solution in the domain
Economists and sociologists have been paying attention to the speed of new products, hoping to establish a mathematical model to describe it and use it to guide production .
Suppose a new product is to be launched on the market and the number of new products sold at
Its solution is:
According to Eq. (5), its first and second derivatives can be obtained:
It is not difficult to see that when
We see that although
This is inflation. It is caused by a short supply. In order to stabilise prices, it is necessary to reduce consumption funds, reduce demand or increase the supply of goods. For example, reducing the number of government officials and redundant employees in enterprises can reduce consumption funds. Or the implementation of measures such as the promotion of commercial housing is to increase the supply of goods, which can play a role in restraining price increases .
Under a certain abnormal economic situation, consumers’ shopping psychology is abnormal. The more the price increases, the more they buy, and the faster the price increases, the more they buy, so demand
The singularity of Eq. (15) is
It can be seen from the phase diagram that the right half-plane is divided into two areas by the ray AC, the trajectory starting from above AC, when
Therefore, it is known from Theorem 1 that there is no closed track, that is, prices will not oscillate periodically and can only tend to a stable value
If the output value fluctuates too much over time, it will cause instability of economic life and social life, and economic adjustment must be made to make it develop in a relatively healthy manner .
Derivation from (20)
Converted into an equivalent system of equations
The singularity is
Remember Δ = (
The general solution of (26) is
In order to be economically stable,
This article mainly makes a simple exploration of the application of differential equations in economic systems from the perspective of mathematics, concretely analyses specific problems in the economic system  and makes corresponding assumptions and simplifications for establishing appropriate differential equation models. The listed equations are solved according to the laws and methods in the field of differential equations. The results obtained were described and analysed, and finally reflected in economic reality .