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Course

Differential Equations

PatrickJMT
Course Lectures
  • Logistic Differential Equation
    Logistic Differential Equation
    Patrick Jones
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    In this module, we explore the logistic differential equation, a crucial model in understanding population growth. This equation represents how populations grow in an environment with limited resources. The key focus is on:

    • The formulation of the logistic differential equation.
    • Finding its solution.
    • Using the solution to predict population sizes over time.

    By the end of the module, you will have a solid grasp of how this equation is applied in real-world scenarios, particularly in biology and ecology.

  • Euler's Method - Another Example #1
    Euler's Method - Another Example #1
    Patrick Jones
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    This module presents another practical example of Euler's Method, a numerical technique used for approximating solutions to differential equations. In this lesson, you will:

    • Understand the principles of Euler's Method.
    • Follow through a detailed example, step-by-step.
    • Recognize the limitations of this method in finding exact solutions.

    By engaging with this module, you will enhance your skills in numerical analysis and gain a clearer understanding of how to apply this method effectively.

  • Euler's Method - Another Example #2
    Euler's Method - Another Example #2
    Patrick Jones
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    In this module, we continue our exploration of Euler's Method by analyzing a second example. This module emphasizes the numerical approximation of solutions to differential equations. You will learn:

    • The step-by-step process of applying Euler's Method.
    • How to interpret the results of your approximations.
    • Common pitfalls to avoid when using this method.

    Completing this module will deepen your understanding of numerical solutions and their practical applications in various fields.

  • Solving a Separable Differential Equation, Another Example #1
    Solving a Separable Differential Equation, Another Example #1
    Patrick Jones
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    This module focuses on solving separable differential equations, a fundamental concept in differential equations. In this example, we will:

    • Identify separable equations and their characteristics.
    • Follow a systematic approach to solve them.
    • Apply integration techniques to find the general solution.

    By the end of this module, you will have a clear understanding of how to tackle separable equations effectively, a skill crucial for advanced mathematical studies.

  • Solving a Separable Differential Equation, Another Example #2
    Solving a Separable Differential Equation, Another Example #2
    Patrick Jones
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    This module continues the exploration of separable differential equations with another example. You will further develop your skills in:

    • Recognizing the structure of separable equations.
    • Applying the separation of variables method.
    • Finding solutions through integration.

    By completing this module, you will enhance your problem-solving abilities and gain confidence in handling various types of differential equations.

  • Solving a Separable Differential Equation, Another Example #3
    Solving a Separable Differential Equation, Another Example #3
    Patrick Jones
    Play

    Here, we tackle a third example of solving separable differential equations, reinforcing the concepts learned in previous modules. In this session, you will:

    • Apply the separation of variables method to another unique equation.
    • Integrate to find the general solution.
    • Discuss the significance of your findings in broader mathematical contexts.

    This module aims to solidify your understanding and prepare you for more complex differential equations in future studies.

  • Solving a Separable Differential Equation, Another Example #4, Initial Condition
    Solving a Separable Differential Equation, Another Example #4, Initial Condition
    Patrick Jones
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    This module introduces a more complex example of solving separable differential equations that includes an initial condition. You will learn how to:

    • Incorporate initial conditions into the solution process.
    • Understand the importance of initial conditions in real-world applications.
    • Complete the solution with specific values for greater accuracy.

    By the end of this module, you will appreciate the role of initial conditions in differential equations and be equipped to handle similar problems confidently.

  • Solving a Separable Differential Equation, Another Example #5, Initial Condition
    Solving a Separable Differential Equation, Another Example #5, Initial Condition
    Patrick Jones
    Play

    Continuing with our exploration of separable differential equations, this module presents another example that also incorporates an initial condition. Key aspects include:

    • Reviewing the steps to solve with initial conditions.
    • Applying your knowledge to find specific solutions.
    • Discussing potential applications of these equations.

    This module emphasizes the practical implications of solving these equations and reinforces your skills in mathematical problem-solving.

  • Basic Differential Equation with an Initial Condition
    Basic Differential Equation with an Initial Condition
    Patrick Jones
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    In this module, we explore a basic differential equation that includes an initial condition. You'll learn how to:

    • Identify and formulate the differential equation.
    • Apply integration techniques to find the general solution.
    • Use the initial condition to solve for the constant of integration.

    This foundational knowledge is essential as you progress in your study of differential equations and prepares you for more advanced topics.

  • Differential Equations - Basic Idea of What It Means to be a Solution
    Differential Equations - Basic Idea of What It Means to be a Solution
    Patrick Jones
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    This module discusses the basic concept of what it means to be a solution to a differential equation. You'll discover:

    • The fundamental definition of a solution in the context of differential equations.
    • How to verify if a given function is a solution.
    • Real-world applications of differential equations in modeling.

    This foundational understanding will provide you with the necessary background for more complex topics in differential equations.

  • First Order Linear Differential Equations / Integrating Factors - Ex 2
    First Order Linear Differential Equations / Integrating Factors - Ex 2
    Patrick Jones
    Play

    In this module, we delve into first-order linear differential equations and the method of integrating factors through a second example. You will learn:

    • The structure of first-order linear differential equations.
    • How to apply integrating factors effectively.
    • Step-by-step guidance through the solution process.

    This module will enhance your understanding of first-order linear equations and prepare you for more advanced topics in differential equations.

  • Change of Variables / Homogeneous Differential Equation - Example 1
    Change of Variables / Homogeneous Differential Equation - Example 1
    Patrick Jones
    Play

    This module focuses on solving homogeneous differential equations using the change of variables technique. In this first example, you will:

    • Learn how to identify homogeneous equations.
    • Understand the process of applying a suitable change of variables.
    • Complete the solution with clear explanations.

    By the end of this module, you will have a solid foundation in using change of variables to solve homogeneous differential equations.

  • Change of Variables / Homogeneous Differential Equation - Example 2
    Change of Variables / Homogeneous Differential Equation - Example 2
    Patrick Jones
    Play

    Continuing with the change of variables technique, this module presents a second example of solving homogeneous differential equations. You will:

    • Reinforce your understanding of homogeneous equations.
    • Apply the change of variables method to a new equation.
    • Analyze the solutions obtained and their implications.

    This module is designed to strengthen your skills and confidence in tackling similar problems in your future studies.

  • Change of Variables / Homogeneous Differential Equation - Example 3
    Change of Variables / Homogeneous Differential Equation - Example 3
    Patrick Jones
    Play

    This module presents a third example of solving homogeneous differential equations using the change of variables method. Key aspects include:

    • Identifying the structure of the differential equation.
    • Applying the change of variables to simplify the equation.
    • Finding and discussing the solutions obtained.

    By the end of this module, you will have a deeper understanding of how to handle homogeneous differential equations effectively.

  • Change of Variables / Homogeneous Differential Equation - Example 4
    Change of Variables / Homogeneous Differential Equation - Example 4
    Patrick Jones
    Play

    In this module, we tackle the fourth example of solving homogeneous differential equations using a change of variables. This example is unique due to its complexity, which includes:

    • Using partial fractions to integrate the resulting equation.
    • Completing the solution step-by-step.
    • Discussing the significance of the method applied.

    By the end of this module, you will be equipped to handle more challenging homogeneous equations confidently.

  • The Inverse Laplace Transform - Example and Important Theorem
    The Inverse Laplace Transform - Example and Important Theorem
    Patrick Jones
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    This module introduces the concept of the inverse Laplace transform, including key definitions and an important theorem. In this session, you will learn:

    • The significance of the inverse Laplace transform in solving differential equations.
    • Step-by-step guidance on finding the inverse Laplace transform of a function.
    • How to apply the theorem in practical scenarios.

    This foundational knowledge will prepare you for more advanced topics involving Laplace transforms in differential equations.

  • Table of Laplace Transforms
    Table of Laplace Transforms
    Patrick Jones
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    In this module, we present a table of Laplace transforms for common functions. This resource is designed to:

    • Provide quick reference material for Laplace transforms.
    • Facilitate your understanding of how these transforms are used in differential equations.
    • Help you identify transforms relevant to your studies.

    This quick reference will be beneficial as you progress through more complex applications of Laplace transforms in your coursework.

  • The Laplace Transform - The Basic Idea of How We Use It
    The Laplace Transform - The Basic Idea of How We Use It
    Patrick Jones
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    This module focuses on the basic idea of how we use the Laplace transform in solving differential equations. Key points include:

    • The concept of transforming a function to simplify the solving process.
    • Understanding the properties of the Laplace transform.
    • Real-world applications of the Laplace transform in engineering and physics.

    By the end of this module, you will have a foundational understanding of the Laplace transform and its significance in engineering and physics.

  • Laplace Transform is a Linear Operator - Proof
    Laplace Transform is a Linear Operator - Proof
    Patrick Jones
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    In this module, we explore the concept that the Laplace transform is a linear operator. You will discover:

    • The definition of a linear operator and its properties.
    • How the linearity of the Laplace transform simplifies solving differential equations.
    • Examples that illustrate the concept in action.

    This foundational knowledge will enhance your understanding of the Laplace transform and its applications in various mathematical contexts.

  • The Laplace Transform, Basic Properties - Definitions and Derivatives
    The Laplace Transform, Basic Properties - Definitions and Derivatives
    Patrick Jones
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    This module introduces the foundational concepts of the Laplace Transform, focusing on its basic properties and definitions. You will explore:

    • The definition of the Laplace Transform
    • Basic properties such as linearity and shifting
    • How to differentiate transforms and their applications

    Understanding these properties is crucial for solving differential equations effectively, as they provide powerful tools for analysis.

  • The Laplace Transform - More Derivatives
    The Laplace Transform - More Derivatives
    Patrick Jones
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    This module builds upon the previous one by delving into more advanced aspects of the Laplace Transform. Key topics include:

    • Higher-order derivatives and their significance
    • Inversion techniques for Laplace Transforms
    • Practical applications in engineering and physics

    By mastering these advanced derivatives, you will enhance your ability to tackle complex differential equations.

  • The Logistic Equation and Models for Population - Example 1, part 1
    The Logistic Equation and Models for Population - Example 1, part 1
    Patrick Jones
    Play

    In this module, we explore the concept of the Logistic Equation through a practical example involving fish population dynamics. You will learn how:

    • Biologists stock a lake and observe population growth
    • The population triples in the first year
    • To derive a formula for population over time using the analytic solution

    This example illustrates the application of the logistic model in real-world scenarios, emphasizing the importance of carrying capacity.

  • The Logistic Equation and Models for Population - Example 1, part 2
    The Logistic Equation and Models for Population - Example 1, part 2
    Patrick Jones
    Play

    This module continues the exploration of the Logistic Equation, focusing on the timeline for fish population growth. Key learning points include:

    • How to calculate the time taken for a population to reach a specific size
    • Application of the logistic model to real-life scenarios
    • Understanding the dynamics of population growth over time

    By analyzing this case, you will gain insights into the practical implications of population models.

  • Power Series Solutions of Differential Equations
    Power Series Solutions of Differential Equations
    Patrick Jones
    Play

    This module introduces Power Series as a method for solving differential equations. You will explore:

    • The concept of power series and their convergence
    • Step-by-step methods for applying power series to differential equations
    • Examples illustrating simple solutions and their effectiveness

    This approach provides a valuable alternative to traditional methods, showcasing the beauty and simplicity of certain solutions.

  • The Logistic Equation and the Analytic Solution
    The Logistic Equation and the Analytic Solution
    Patrick Jones
    Play

    In this module, we will find the analytic solution to the logistic differential equation. Key topics include:

    • The derivation of the logistic equation solution
    • Understanding the implications of the solution in population modeling
    • Applications of the analytic solution in various fields

    By the end, you will appreciate the relevance of analytic solutions in understanding population dynamics.

  • Exact Differential Equations
    Exact Differential Equations
    Patrick Jones
    Play

    This module focuses on Exact Differential Equations, a key concept in solving specific types of differential equations. You will learn about:

    • The definition and criteria for exactness in differential equations
    • Step-by-step methods to solve exact differential equations
    • An example demonstrating how to apply these concepts effectively

    By understanding exactness, you will enhance your problem-solving skills in differential equations.

  • First Order Linear Differential Equations
    First Order Linear Differential Equations
    Patrick Jones
    Play

    In this module, we will look at First Order Linear Differential Equations and their significance in mathematical modeling. Key learning points include:

    • The standard form of first order linear equations
    • Methods for finding solutions, including integrating factors
    • Applications in various fields such as physics and engineering

    By mastering these techniques, you will be equipped to tackle a wide range of real-world problems.

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