SOLVING ECONOMIC PROBLEMS USING DIFFERENTIAL EQUATIONS

Аннотация

This article highlights the role and importance of differential equations in modeling modern economic processes. It analyzes time-dependent economic indicators such as population growth, price dynamics, investment efficiency, and changes in production volume using mathematical models. The application of first-order differential equations is demonstrated using real examples such as the Solow economic growth model, the Malthus population model, and the price equilibrium model. In addition, the process of accumulating interest on a bank deposit savings strategy is modeled using a differential equation. The graphs presented in the article visually illustrate the changes in economic indicators over time. At the end of the study, the use of differential equations as an effective tool in economic analysis and forecasting is emphasized. Keywords: differential equation, economic modeling, Solow model, Malthus model, price dynamics, population growth, capital changes, linear differential equation, economic forecast, mathematical model. Time-varying processes are important in the analysis of modern economics. For example, many economic phenomena, such as population growth, investment efficiency, price dynamics, and changes in production, can be transformed into mathematical models using differential equations. Adifferential equation is an equation that contains an unknown function and its derivatives. dy(t) = f (t,y) dt here y ( t )- economic quantity (e.g. price, population,production volume), t - time. In economics, these functions are usually time-dependent, for example: price changes, output, population,inflation rate, interest rate, etc. Solving economic problems usingdifferential equations isa method of analyzing thechanges in economic systems and processes over time by mathematically modeling them. This method is used in many economic models and forecasts. Below are themain concepts, areas ofapplication andexamplesonthis topic. Applications ineconomic modeling page1813 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue05,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Solow economic growth model. (Solow is a famous model of economic growth theory, developed byAmerican economist Robert Solow.) This model represents output (Y) using capital (K), labor (L), and technology (A). Capital changes are modeled using adifferential equation: dK(t) = sY(t)-dK(t) dt here: s - savings rate, d- capital depreciation ratio, Y(t)= A(t)K(t)aL(t)1-a –production function (in Cobb-Douglas form). Populationgrowth model (Malthus model) dP(t) = rP(t) dt here P(t)- population over time, r - growth rate t - time(in years). Solution: P(t)= Pert 0 Thismodel shows exponential populationgrowth. Example: Acity hada population of100,000in 2020.Thepopulation is growing at a rate of3% peryear.Calculate thepopulation ofthis cityby2025. Given:P =100000 0 r =0,03 (i.e. 3%) t=5year (2020-2025) Solution: P(5)=100000 e0,035 =100000 e0,15 P(5) =100000 1,1618=116180 If thegrowth rate does not change, thepopulation willreach 116,180in 5years. Modelingprice changes Assume that price is time-dependent. Based on supply and demand, the following equation can beconstructed: dp(t) =a(D(p)- S(p)) dt hereis D(p) thedemand function, S(p)thesupply function. Priceandsupply-demand model:The essence of theproblem. If demand increases, the price increases, and if supply increases, the price decreases. This process needs to bemodeled. Mathematical model: dp =a(D(p)- S(p)) dt Let's assume: D ( p ) =a -bp (demand), S ( p ) =c +dp (offer). Then: page1814 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue05,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai dp =a[ (a-bp)-(c+ dp) ]=a( a-c-(b+ d)p ) dt Thisis afirst-order lineardifferential equation. Solution: dp +a(b+ d)p =a(a-c) dt Solvedbyintegration: a-c p(t) =Ce-a(b+d)t + b+d Thismodel shows that theprice approaches theequilibrium price overtime. The following example mathematically models a savings strategy based on a differential equation: A woman wants to save $60,000 for her child's future study abroad in 5 years. She uses a deposit in a bank with an annual interest rate of 8%. The bank's interest is calculated continuously. The woman also deposits an additional $2,000 into the deposit account each year. That is, the total amount of money in the bank increases not only due to interest, but also due to the amount added annually. What should be the initial amount of money in the bank now (at time ) so that thetotalamount in theaccount after 5years is $60,000? ( ) y t t theamount ofmoney in thebank account at thetime(in dollars), t time(year), r =0,08 annual interest rate(the interest rate is expressed as adecimal), Theadditional amount to beadded eachyear is 2000. dy =0,08y(t)+ 2000 This equation is a first-order linear differ