1.3 Mathematical Modeling with Differential Equations
4 min read•Last Updated on August 6, 2024
Mathematical modeling with differential equations is a powerful tool for describing real-world phenomena. It allows us to represent complex systems using mathematical language, helping us understand and predict their behavior over time.
In this section, we explore various models, including growth and decay, physical systems, and compartmental models. These applications showcase how differential equations can be used to solve practical problems across different fields of study.
Growth and Decay Models
Exponential Growth and Decay
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Exponential growth and decay models describe situations where the rate of change of a quantity is proportional to the current amount of that quantity
The general form of an exponential growth or decay model is dtdy=ky, where k is the growth or decay rate constant
If k>0, the model represents exponential growth (population growth, compound interest)
If k<0, the model represents exponential decay (radioactive decay, cooling of an object)
The solution to an exponential growth or decay model is given by y(t)=y0ekt, where y0 is the initial value of the quantity at time t=0
Logistic Growth
Logistic growth models describe situations where a population grows exponentially until it reaches a carrying capacity, at which point the growth rate slows down and eventually stabilizes
The logistic growth model is given by dtdy=ky(1−Ly), where k is the growth rate constant and L is the carrying capacity
The term (1−Ly) represents the effect of limited resources on the growth rate as the population approaches the carrying capacity
The solution to the logistic growth model is y(t)=1+(y0L−1)e−ktL, where y0 is the initial population size
Population Dynamics
Population dynamics models describe how populations of different species interact with each other and their environment over time
These models can include factors such as birth rates, death rates, competition, predation, and resource availability
Examples of population dynamics models include predator-prey models (Lotka-Volterra equations) and competition models (Competitive Lotka-Volterra equations)
Physical Systems
Newton's Law of Cooling
Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its temperature and the ambient temperature
The model is given by dtdT=−k(T−Ta), where T is the temperature of the object, Ta is the ambient temperature, and k is the cooling rate constant
The solution to Newton's law of cooling is T(t)=Ta+(T0−Ta)e−kt, where T0 is the initial temperature of the object
Mechanical Systems
Mechanical systems, such as spring-mass systems and pendulums, can be modeled using differential equations
For example, a spring-mass system can be modeled by mdt2d2x+cdtdx+kx=F(t), where m is the mass, c is the damping coefficient, k is the spring constant, and F(t) is the external force applied to the system
The solutions to these models describe the motion of the mechanical system over time (oscillations, damping, resonance)
Electrical Circuits
Electrical circuits can be modeled using differential equations based on Kirchhoff's laws and the relationships between voltage, current, and circuit elements (resistors, capacitors, inductors)
For example, an RC circuit can be modeled by RCdtdV+V=Vs(t), where R is the resistance, C is the capacitance, V is the voltage across the capacitor, and Vs(t) is the source voltage
The solutions to these models describe the behavior of the electrical circuit over time (charging, discharging, transient response)
Compartmental Models
Compartment Models
Compartment models describe the flow of a substance between different compartments or pools within a system
These models are often used in pharmacokinetics (drug absorption, distribution, metabolism, and elimination), epidemiology (spread of diseases), and ecology (nutrient cycling)
The general form of a compartment model is a system of first-order linear differential equations, with each equation representing the rate of change of the amount of substance in a particular compartment
For example, a two-compartment model can be represented by dtdA1=−k12A1+k21A2 and dtdA2=k12A1−k21A2, where A1 and A2 are the amounts of substance in compartments 1 and 2, and k12 and k21 are the transfer rate constants between the compartments
Mixing Problems
Mixing problems involve the mixing of two or more substances in a tank or container, with the goal of determining the concentration of the substances over time
These problems can be modeled using first-order linear differential equations, with the rate of change of the amount of substance in the tank depending on the inflow and outflow rates and the volume of the tank
For example, if a tank with volume V contains a mixture with concentration c(t), and a solution with concentration cin is flowing in at a rate rin while the mixture is flowing out at a rate rout, the model is given by Vdtdc=rin(cin−c)−routc