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.

#### Keywords

- differential equation
- model
- general solution
- economy
- price

#### MSC 2010

- 34A34

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 [1]. 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 [2] 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

^{*}, then the system of equations

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 [3].

Suppose a new product is to be launched on the market and the number of new products sold at

Solution (2):

If _{0} has been put into use through hard work, then the growth of

Its solution is:

The curve

According to Eq. (5), its first and second derivatives can be obtained:

It is not difficult to see that when ^{′}^{−Mrt} = 1 is derived from ^{″}_{0}) = 0, and
_{0}, ^{″}_{0}) ^{′}^{′}_{0}, ^{″}_{0}) ^{′}_{0}). The product was the most popular, followed by (_{0}), and the sales rate began to decline. From this survey by many economic experts and product salespeople, the sales curve of many new products is quite close to the logistic curve. Therefore, it is possible to conduct an in-depth analysis of the sales curve characteristics of new products and conclude that new products should be produced in a small amount in the early stage of sales, and at the same time, publicity and advertising should be strengthened. When the new product consumers reach 20–80%, from time to time, new products can be produced in large quantities; when the consumers of new products exceed 80%, then the company should choose the right time to switch production in order to achieve better interests of the company [4].

Price

Among them,

Among them,

And 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 [5].

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

Finished up:

Let

The singularity of Eq. (15) is

_{1} and _{2} are real numbers with different signs, and the singularity is a saddle point, as shown in Figure 2.

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 _{2} _{2} _{2} _{2}

_{1} and _{2} are two negative real numbers or conjugate complex numbers with negative real parts,

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 [7].

Suppose

Then

Derivation from (20)

From (21)

Converted into an equivalent system of equations

The singularity is

Remember Δ = (^{2} − 4

If _{1}, _{2} are pure imaginary numbers, then (_{0}_{0}) is the centre, as shown in Figure 3. At this time,

The general solution of (26) is

_{1} and

If _{1}, _{2} are all positive real numbers, and (_{0}_{0}) is an unstable node, as shown in Figure 4.

If _{1} = _{2} _{0}_{0}) is an unstable node, as shown in Figure 5.

If _{1}, _{2} is a conjugate complex number with positive real part, and (_{0}_{0}) is an unstable focus, as shown in Figure 6.

If _{1}, _{2} are all negative real numbers, and (_{0}_{0}) is a stable node, as shown in Figure 7.

If _{1} = _{2} _{0}_{0}) is a stable node, see Figure 8.

If _{1}, _{2} is a conjugate complex number with negative real part, and (_{0}_{0}) is a stable focus, as shown in Figure 9.

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 [9] 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 [10].

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