Modelling with First-order Differential Equations

Modelling with first-order differential equations is a key component of further mathematics that has vast real-world applications across various fields such as population dynamics, finance, physics, and engineering. Understanding the concept of mathematical modelling and the role of ordinary differential equations in modelling is essential for successfully applying these techniques. In this text, you will explore different examples and applications, delve into interest problems, and learn about the various techniques for solving first-order ordinary differential equations. With a step-by-step approach to forming mathematical models, solving equations, and interpreting results, you will gain the necessary knowledge and skills to apply these methods in real-world situations. Discover the fascinating world of modelling with first-order differential equations and broaden your mathematical understanding and expertise.

Modelling with First-order Differential Equations Modelling with First-order Differential Equations

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Contents
Table of contents

    Understanding Modelling with First-order Differential Equations

    The concept of mathematical modelling refers to the process of creating a mathematical structure that represents a real-life situation. Often, such models are designed to analyse complex systems and make predictions or solve problems. There are numerous reasons to employ mathematical modelling, including:

    • Analysing the behaviour of a system under different conditions
    • Predicting the future behaviour of a system
    • Finding the best possible solution for a given problem

    Mathematical models can vary in complexity, from simple linear relationships to highly intricate structures involving multiple variables and equations.

    While constructing a mathematical model, one should consider the following steps:

    1. Identify the problem and variables
    2. Formulate assumptions and simplifications
    3. Derive the mathematical relationships
    4. Solve the equations and interpret the results
    5. Validate the model by comparing predictions with real data

    The role of Ordinary Differential Equations in Modelling

    In many modelling scenarios, ordinary differential equations (ODEs) play a vital role. ODEs are equations that contain one or more derivatives of a dependent variable with respect to an independent variable.

    An ODE can be expressed as \(\frac{dy}{dx} = f(x, y)\), where \(f\) is a function of \(x\) and \(y\), and \(y\) is the dependent variable which is differentiated with respect to the independent variable \(x\).

    First-order differential equations are a particular class of ODEs where the highest derivative is of the first order. Modelling with first-order differential equations is common in various scientific and engineering fields. Some examples include:

    • Population dynamics
    • Chemical reaction kinetics
    • Electrical circuits
    • Heat conduction

    An example of a first-order differential equation is the exponential growth model, which can be represented as \(\frac{dP}{dt} = kP\), where \(P\) is the population, \(t\) is time, and \(k\) is the constant growth rate.

    In modelling with first-order differential equations, specific terminologies and concepts are essential to comprehend:

    Initial value problem (IVP)A problem in which the solution of a first-order differential equation must satisfy an initial condition
    Separation of variablesA technique for solving first-order ODEs by isolating the dependent and independent variables on opposite sides of the equation
    Exact differential equationA first-order ODE in which a specific condition is met, allowing it to be expressed as the differential of a single function
    Linear first-order ODEAn ODE in the form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x)\) and \(Q(x)\) are functions of \(x\)

    Effective modelling with first-order differential equations requires an understanding of these concepts and a solid foundation in calculus, as solution techniques often involve integration and differentiation.

    Examples of Modelling with First-order Differential Equations

    First-order differential equations can be utilised to model numerous real-world scenarios across various domains, such as population dynamics, business and finance, and physics and engineering. Through their application, these models help to analyse, predict, and optimise complex systems.

    Applications in Population Dynamics

    Population dynamics encompass the study of how populations of living organisms change over time. First-order differential equations are an effective tool in modelling these changes for scenarios such as:

    1. Exponential growth: This model represents a population growing at a constant rate, characterised by the equation \(\frac{dP}{dt} = kP\), where \(P\) is the population, \(t\) is time, and \(k\) is the growth rate.
    2. Logistic growth: In this model, a population grows until it reaches a carrying capacity, usually due to limited resources. The logistic growth equation is \(\frac{dP}{dt} = kP\left(1 - \frac{P}{K}\right)\), where \(P\) is the population, \(t\) is time, \(k\) is the intrinsic growth rate, and \(K\) is the carrying capacity.
    3. Compartmental models: These models divide a population into distinct groups or compartments based on criteria such as age or susceptibility to a disease. A well-known example is the SIR model, used to study the spread of infectious diseases. Here, the population is separated into Susceptible (S), Infected (I), and Recovered (R), with first-order ODEs defining the transitions between these groups.

    Modelling in Business and Finance

    First-order differential equations can also be employed to model various business and financial scenarios. Some examples include:

    1. Compound interest: The continuous compounding of interest can be modelled using a first-order ODE, represented as \(\frac{dA}{dt} = rA\), where \(A\) is the account balance, \(t\) is time, and \(r\) is the annual interest rate.
    2. Inventory management: In this context, first-order ODEs can be used to model the rate of change in inventory levels over time due to factors like sales and replenishment. An example ODE for this problem could be \(\frac{dI}{dt} = R(t) - S(t)\), where \(I\) denotes the inventory, \(R(t)\) represents the replenishment rate, and \(S(t)\) is the sales rate.
    3. Economic growth: The Solow growth model, widely used in macroeconomics, describes the determinants of economic growth through a series of first-order ODEs involving variables such as capital, labour, and technology.

    Examples in Physics and Engineering

    Physics and engineering offer countless opportunities for applying first-order differential equation models, with several examples listed below:

    1. Newton's Law of Cooling: This law models the rate at which an object cools in relation to the surrounding medium's temperature. The associated ODE is \(\frac{dT}{dt} = k(T_{s} - T)\), where \(T\) denotes the object's temperature, \(T_{s}\) represents the surrounding medium's temperature, \(t\) is time, and \(k\) is the cooling constant.
    2. Electrical circuits: Modelling the behaviour of circuits, particularly those involving resistors and capacitors, often requires first-order ODEs. The equation for a resistor-capacitor (RC) circuit is \(\frac{dq}{dt} = -\frac{1}{RC}(q - q_{0})\), where \(q\) is the charge on the capacitor, \(t\) is time, \(R\) symbolises resistance, \(C\) signifies capacitance, and \(q_{0}\) denotes the initial charge.
    3. Simple harmonic motion: Forms of this motion, such as a mass-spring system, can be modelled using first-order ODEs. In this case, the equation is \(\frac{dv}{dt} = -\frac{k}{m}x\), where \(v\) is the velocity of the mass, \(t\) is time, \(k\) is the spring constant, \(m\) refers to the mass, and \(x\) signifies the displacement from the equilibrium position.

    Modelling with first-order differential equations is a versatile and powerful approach for exploring a plethora of real-world problems across numerous domains.

    Exploring Modelling with First-order Differential Equations in Interest

    When it comes to modelling interest problems, first-order differential equations can be instrumental in capturing their dynamics and offering insights into various financial scenarios. Interest problems are a critical aspect of finance and economics, and untangling their complexities using first-order differential equations offers practical solutions and valuable predictions.

    Applications of First-order Differential Equations in Interest problems

    First-order differential equations can be applied successfully to various interest problems, encompassing simple interest, compound interest, and continuous compounding. Their role in these applications cannot be overstated, as first-order differential equations provide a mathematical framework for calculating interest, projecting growth, and optimising investments.

    Simple Interest

    Simple interest refers to a fixed percentage of the principal amount that accrues periodically, usually annually. It is the most basic form of interest and can be modelled using a first-order linear equation. The formula for calculating simple interest is given as \(I = P \cdot r \cdot t\), where \(I\) represents the accumulated interest, \(P\) is the initial principal amount, \(r\) stands for the interest rate, and \(t\) denotes the time period (usually in years).

    Although simple interest is not modelled using a first-order differential equation, it provides the foundation for understanding more complex interest problems that require first-order differential equations, such as compound and continuously compounded interest.

    Compound Interest

    Compound interest is a more intricate form of interest where the interest earned is added to the principal, resulting in interest being calculated not just on the original amount but also on the accrued interest. First-order differential equations can be employed to model compound interest, as they capture the changing dynamics of the interest rate over time. The formula for compound interest is given as \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) signifies the accumulated amount, \(P\) represents the initial principal, \(r\) is the interest rate, \(n\) denotes the number of compounding periods within a year, and \(t\) stands for the number of years involved.

    First-order differential equations can be applied to compound interest problems by examining the rate of change of the accumulated amount with respect to time (\(\frac{dA}{dt}\)). In this case, one can derive the equation \(\frac{dA}{dt} = r A\), where \(A\) denotes the accumulated amount, \(t\) is time, and \(r\) signifies the interest rate.

    An example of a compound interest problem modelled using a first-order differential equation could be a savings account with an initial deposit of £1,000, an annual interest rate of 5%, and semi-annual compounding. Using the compound interest equation and solving the associated first-order differential equation, one can calculate the accumulated amount after a certain number of years.

    Continuous Compounding

    Continuous compounding takes the concept of compound interest a step further, considering the situation where compounding takes place an infinite number of times per year. This results in a unique modelling framework that employs first-order differential equations. The equation for continuously compounded interest is given as \(A = Pe^{rt}\), where \(A\) symbolises the accumulated amount, \(P\) denotes the initial principal, \(r\) represents the interest rate, and \(t\) refers to the time involved, usually expressed in years.

    When modelling continuous compounding with first-order differential equations, the rate of change of the accumulated amount with respect to time (\(\frac{dA}{dt}\)) is evaluated. The equation then takes the form \(\frac{dA}{dt} = rA\), where \(A\) indicates the accumulated amount, \(t\) signifies time, and \(r\) is the interest rate. Solving this equation ultimately reveals the exponential growth of the accumulated amount as a function of time.

    An example of continuous compounding modelled with a first-order differential equation could involve a £1,000 initial deposit into an account with an annual interest rate of 5% and continuous compounding. By solving the first-order differential equation and applying the continuous compounding equation, one can accurately project the accumulated amount after a specified number of years.

    First-order differential equations offer valuable insights in modelling interest problems and provide a robust mathematical framework to examine the intricacies of simple, compounded, and continuously compounded interest cases.

    Steps in Modelling with First-order Ordinary Differential Equations

    Forming a mathematical model

    To construct an effective mathematical model with first-order ordinary differential equations (ODEs), it is essential to carefully follow these steps:

    1. Identify the problem and variables: Begin by understanding the real-world scenario you want to model and determine the relevant independent and dependent variables.
    2. Formulate assumptions and simplifications: To simplify the model, make reasonable assumptions and reduce complexity. These assumptions may pertain to the behaviour of the system or may involve neglecting small effects that are considered insignificant.
    3. Derive the mathematical relationships: Establish relationships between the variables based on the problem's context and your understanding of the situation. Transform these relationships into a system of first-order ODEs, explicitly describing the problem's rate of change.

    During the formation of a mathematical model, it is vital to examine the problem, identify critical features, and derive appropriate mathematical expressions that accurately describe the situation.

    Solving the First-order Ordinary Differential Equation

    Once the mathematical model has been formulated, the next step is to solve the first-order ODE. Several methods are available depending on the equation's form, such as separation of variables, integrating factors, and exact equations, among others. The critical aspect here is to apply the most suitable technique. Some popular solution methods include:

    1. Separation of Variables: In this technique, the dependent and independent variables are isolated on opposite sides of the equation, allowing for direct integration. The method is most applicable when the equation can be expressed as \(\frac{dy}{dx} = g(y)h(x)\).
    2. Integrating Factors: Used for solving linear first-order ODEs in the form \(\frac{dy}{dx} + P(x)y = Q(x)\), an integrating factor is a function that simplifies the equation, resulting in an exact differential which can then be integrated directly.
    3. Exact Equations: If the ODE fulfils a specific condition, it can be restructured as the differential of a single function. To determine whether an equation is exact, the necessary test should be performed, and if the test is passed, further steps are taken to solve the ODE.

    Additionally, various numerical methods, such as Euler's method and Runge-Kutta methods, can be employed to approximate solutions if the first-order ODE cannot be solved analytically.

    Interpreting the results and determining accuracy

    After solving the first-order ODE, the next step is to interpret the results and assess their accuracy. This phase involves:

    1. Interpreting Solutions: Analyse the mathematical solutions obtained and relate them back to the real-world problem. The solution should provide insights into the behaviour of the system and, if possible, bring forth new predictions or solutions for the problem at hand.
    2. Comparing with Real Data: Validate the model by comparing its predictions to existing real-world data. This process helps identify discrepancies and improve the accuracy of the model before applying it to unexplored scenarios.
    3. Evaluating Precision: Determine the level of accuracy of the model's solutions to real data. Examine the assumptions made when forming the model and, if necessary, refine the model to reduce potential errors.

    In conclusion, modelling with first-order differential equations involves systematically formulating a mathematical model, solving the ODE, and interpreting the results while ensuring their accuracy. With a well-crafted model and thorough analysis, valuable insights into complex systems can be obtained, leading to optimised decisions and effective problem-solving.

    Techniques for Solving First-order Ordinary Differential Equations

    Solving first-order ordinary differential equations (ODEs) can be a crucial aspect of modelling real-life scenarios and understanding complex systems. Various techniques are available to solve these equations, based on their structures and characteristics. This section highlights some of the most common methods such as Separable Equations, Exact Equations, Integrating Factor Method, and Linear First-order Differential Equations.

    Separable Equations

    Separable equations are a type of first-order ODE wherein the dependent and independent variables can be isolated on opposite sides of the equation. To solve separable equations, follow these steps:

    1. Rewrite the ODE in the form \(\frac{dy}{dx} = g(y)h(x)\)
    2. Separate the variables: \(\frac{dy}{g(y)} = h(x) dx\)
    3. Integrate both sides with respect to their respective variables
    4. Solve for the dependent variable \(y\) as a function of the independent variable \(x\) or obtain an implicit solution

    Consider the ODE \(\frac{dy}{dx} = 2xy\). By separating the variables, we obtain \(\frac{dy}{y} = 2x dx\). Integrating both sides results in \(\ln|y| = x^2 + C\), where \(C\) is the integration constant. Finally, solving for \(y\), we get \(y = ke^{x^2}\), where \(k = e^C\).

    Exact Equations

    An exact equation is a first-order ODE that can be expressed as a total differential, permitting the direct determination of a potential function. To solve exact equations, follow these steps:

    1. Write the ODE in the form \(M(x, y) dx + N(x, y) dy = 0\)
    2. Determine if the equation is exact by verifying if \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\)
    3. Find a potential function \(\psi(x, y)\) such that \(\frac{\partial\psi}{\partial x} = M(x, y)\) and \(\frac{\partial\psi}{\partial y} = N(x, y)\)
    4. Obtain the solution by setting \(\psi(x, y) = C\) for a constant \(C\)

    Consider the ODE \((2x + y^3) dx + (3y^2 + 4x) dy = 0\). First, check for exactness by calculating \(\frac{\partial(2x + y^3)}{\partial y} = 3y^2\) and \(\frac{\partial(3y^2 + 4x)}{\partial x} = 3y^2\). Since these partial derivatives are equal, the equation is exact. Next, find the potential function \(\psi(x, y)\) by integrating \(M\) with respect to \(x\) and \(N\) with respect to \(y\), resulting in \(\psi(x, y) = x^2 + y^3 + 2xy + C\). Finally, the solution is given by \(\psi(x, y) = x^2 + y^3 + 2xy = C\), for a constant \(C\).

    Integrating Factor Method

    The integrating factor method is a technique for solving linear first-order ODEs, which are of the form \(\frac{dy}{dx} + P(x) y = Q(x)\). The steps to apply this method are as follows:

    1. Determine an integrating factor (IF), which is given by \(\mu(x) = e^{\int P(x) dx}\)
    2. Multiply the original ODE by the integrating factor: \(\mu(x)\frac{dy}{dx} + \mu(x)P(x)y = \mu(x)Q(x)\)
    3. Notice that the left-hand side (\(\mu(x)\frac{dy}{dx} + \mu(x)P(x)y\)) is now an exact differential of the form \(\frac{d}{dx}[y\mu(x)]\). Thus, integrate both sides with respect to \(x\)
    4. Solve for the dependent variable \(y\) as a function of the independent variable \(x\)

    Consider the ODE \(\frac{dy}{dx} - 2xy = e^{x^2}\). First, determine the integrating factor as \(\mu(x) = e^{\int -2x dx} = e^{-x^2}\). Multiply the ODE by the integrating factor, which results in \(e^{-x^2}\frac{dy}{dx} - 2xe^{-x^2}y = 1\). The left-hand side is now the exact differential, \(\frac{d}{dx}[ye^{-x^2}] = 1\). Integrating both sides and solving for \(y\), we obtain \(y = e^{x^2}\left(\int e^{-x^2} dx + C\right)\), where \(C\) is the integration constant.

    Linear First-order Differential Equations

    Linear first-order ODEs have the form \(\frac{dy}{dx} + P(x) y = Q(x)\) and can be solved using the integrating factor method, as previously described. These ODEs are characterised by the absence of nonlinear terms (such as \(y^2\), \(\sin(y)\), or \(ye^y\)) in the equation. By applying the integrating factor technique, a general expression for the solution can be derived, and specific solutions can be obtained by applying initial conditions.

    Note that when \(Q(x) = 0\), the linear first-order ODE becomes homogeneous, making it separable and solvable using the separation of variables technique.

    In summary, an array of techniques, such as separable equations, exact equations, integrating factor method, and linear first-order differential equations, are available for solving first-order ODEs. The choice of technique depends on the form of the equation and the context of the problem at hand. Familiarity with these methods is crucial for effectively modelling and analysing real-world situations using first-order differential equations.

    Real-world Applications of Modelling with First-order Differential Equations

    Ordinary differential equations (ODEs) are versatile mathematical tools used to model a wide range of real-world phenomena. First-order ODEs, in particular, are applied in numerous fields, including biology, chemistry, physics, and engineering. The modelling process requires a careful evaluation of each problem's unique context, an understanding of essential variables, and the formulation of first-order ODEs to represent the system's behaviour over time accurately.

    Population growth and decay

    First-order differential equations are instrumental in modelling population growth and decay in fields such as ecology and epidemiology. Some examples where first-order ODEs are used in population dynamics include:

    • Exponential growth and decay models: These models represent situations where the population grows or decays at a constant rate over time. The first-order ODE for this model takes the form \(\frac{dP}{dt} = kP\), where \(P\) denotes the population, \(t\) is time, and \(k\) is the growth or decay rate.
    • Logistic growth models: Often used for modelling limited resources or carrying capacity, logistic growth models characterise the population growth that slows down as it approaches the maximum sustainable population. The logistic growth ODE is represented as \(\frac{dP}{dt} = kP\left(1 - \frac{P}{K}\right)\), where \(P\) is the population, \(t\) is time, \(k\) is the intrinsic growth rate, and \(K\) is the carrying capacity.

    Chemical reactions and drug dosage

    First-order ODEs are also employed in chemical kinetics and pharmacokinetics to model reaction rates and drug dosage. Applications include:

    • Chemical reaction rates: In chemistry, first-order ODEs can model reaction kinetics involving simple chemical reactions. For instance, the reaction rate ODE for a single reactant can be represented as \(\frac{d[A]}{dt} = -k[A]\), where \([A]\) denotes the concentration of the reactant, \(t\) is time, and \(k\) is the reaction rate constant.
    • Drug dosage and elimination: In pharmacokinetics, first-order ODEs can describe the drug dosage's absorption and elimination within the body. An example ODE for this scenario could be \(\frac{dC}{dt} = k_a D - k_e C\), where \(C\) is the drug concentration, \(t\) is time, \(D\) denotes the drug dose, and \(k_a\) and \(k_e\) represent the absorption and elimination rate constants, respectively.

    Fluid flow and heat transfer

    First-order ODEs find applications in modelling fluid flow and heat transfer in physics and engineering. Some notable examples include:

    • Torque and speed in rotating systems: First-order ODEs can describe the relationship between torque, inertia, and speed in rotating mechanical systems. An example ODE is \(\frac{d\omega}{dt} = \frac{T}{J}\), where \(\omega\) represents the angular speed, \(t\) is time, \(T\) denotes the torque, and \(J\) symbolises the moment of inertia.
    • Heat conduction and diffusion: First-order ODEs can model heat conduction and diffusion processes in one-dimensional systems, such as rod-shaped structures. The equation for this can be \(\frac{d^2T}{dx^2} = \frac{1}{\alpha}\frac{dT}{dt}\), where \(T\) signifies temperature, \(x\) is the spatial coordinate, \(t\) is time, and \(\alpha\) denotes the thermal diffusivity.

    In summary, first-order ODEs offer a powerful approach for modelling and analysing various real-world problems in diverse domains. By applying these equations to specific scenarios, invaluable insights into system behaviour and solutions to complex challenges can be achieved.

    Modelling with First-order Differential Equations - Key takeaways

    • Modelling with first-order differential equations: widely used in population dynamics, finance, physics, and engineering.

    • Mathematical modelling: process of creating a mathematical structure representing real-life situations for analysis and problem-solving.

    • First-order ordinary differential equations (ODEs): contains one or more derivatives of a dependent variable with respect to an independent variable.

    • Examples of real-world applications: population growth, chemical reactions, fluid flow and heat transfer, and interest-related problems like continuous compounding.

    • Techniques for solving first-order ODEs: Separable Equations, Exact Equations, Integrating Factor Method, and Linear First-order Differential Equations.

    Modelling with First-order Differential Equations Modelling with First-order Differential Equations
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    Frequently Asked Questions about Modelling with First-order Differential Equations
    What can be modelled with differential equations?
    Differential equations can model various real-world phenomena, including population growth, radioactive decay, heat transfer, fluid dynamics, and predator-prey interactions. These equations describe the relationship between a function and its rate of change, allowing us to understand and predict the behaviour of complex systems.
    What is the solution to the first-order differential equation?
    The solution of a first-order differential equation is a function or a family of functions that satisfy the given equation. It represents the general behaviour of the dependent variable in relation to the independent variable and can be found using various methods, including separation of variables, integrating factors, or substituting known solutions.
    When modelling a difference equation, which of the following is the first step?
    The first step when modelling a difference equation is formulating the problem by identifying the variable(s) of interest and their rate of change over time, often represented as a first-order differential equation.
    What are the applications of first-order differential equations?
    First-order differential equations have various applications in areas such as population growth modelling, drug concentration in the body, cooling/heating of objects, electric circuits analysis, and fluid dynamics problems. They help quantify rates of change and allow predictions in diverse scientific and engineering fields.
    What is first-order modelling?
    First-order modelling is a mathematical approach used to describe the behaviour of a system or process by utilising first-order differential equations. These equations involve variables and their derivatives with respect to a single independent variable, usually time. Such modelling is essential for analysing dynamic systems and predicting their future behaviour based on initial conditions or input variables.

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