This course on Differential Equations is designed to provide students with a thorough understanding of the fundamental concepts and techniques used to solve differential equations. Throughout the course, learners will engage with a variety of topics that are crucial for mastering this area of mathematics.
The course is structured around the following key themes:
By the end of this course, students will have developed the skills necessary to solve a variety of differential equations and apply these techniques to real-world problems, particularly in fields like biology and engineering. Join us to deepen your understanding and enhance your mathematical skills!
This module covers the logistic differential equation, a vital concept in modeling population growth. We will explore the equation's solution and how it helps in predicting when a population will reach a specified size.
In this module, we delve into Euler's Method as a practical technique for approximating solutions to differential equations. Through a hands-on example, students will learn how to implement this numerical method effectively.
Continuing with Euler's Method, this module presents another example demonstrating the process of approximating solutions to a differential equation. Students will gain further insight into the practical applications of this method.
This module focuses on solving separable differential equations, emphasizing the method's effectiveness through a detailed example. Students will learn how to separate variables and integrate to find solutions.
In this module, we provide another example of solving a separable differential equation. This continuation helps reinforce the previous concepts, ensuring students are comfortable with the method.
This module presents yet another example of solving a separable differential equation. Students will deepen their understanding of this essential technique through guided practice and detailed explanations.
This module explores solving a separable differential equation that includes an initial condition. Students will learn how to apply initial conditions to find specific solutions to differential equations.
Continuing with the theme of initial conditions, this module offers another example of solving a separable differential equation with an initial condition. This reinforces the application of initial conditions in mathematical modeling.
This module focuses on solving a basic differential equation with an initial condition. Students will practice finding the constant of integration (C) and solidify their understanding of foundational concepts in differential equations.
In this module, we discuss the fundamental idea of what it means to be a solution of a differential equation. Students will learn how to verify solutions by taking derivatives and substituting them back into the equation.
This module introduces first-order linear differential equations and the method of integrating factors. Students will learn how to apply this technique to find solutions for various equations.
This module demonstrates how to solve homogeneous differential equations by employing a change of variables. Students will learn the steps involved in transforming equations to find solutions effectively.
Continuing with the change of variables approach, this module provides another example of solving a homogeneous differential equation. This reinforces the method and expands students' understanding of the topic.
This module presents a third example of solving a homogeneous differential equation through a change of variables. Students will practice the method and gain confidence in their problem-solving abilities.
This module highlights a fourth example of solving a homogeneous differential equation using a change of variables. The complexity of this example will challenge students, as it incorporates partial fractions for integration.
This module introduces the inverse Laplace transform, discussing its significance and providing an important theorem. Students will learn how to find the inverse Laplace transform of a function through practical examples.
In this module, we present a table of Laplace transforms for common functions. This quick reference will aid students in future calculations and serve as a valuable resource throughout the course.
This module provides an overview of the basic idea behind the Laplace transform and how it is utilized in solving differential equations. Students will grasp the foundational concepts that underpin this powerful tool.
This module delves deeper into the properties of the Laplace transform, focusing on derivatives. Students will learn how to apply these properties to simplify the solving process for differential equations.
This module discusses the basic properties of the Laplace transform, providing definitions and insights necessary for understanding its applications in solving differential equations. Students will benefit from this foundational knowledge.
This module continues the exploration of the Laplace transform, specifically focusing on more derivatives and their implications for solving differential equations. Students will gain a deeper understanding of this mathematical tool.
This module presents an example of the logistic equation in population modeling. Through a real-world scenario involving fish population in a lake, students will learn to derive a formula for population over time.
This module continues with the logistic equation, focusing on how long it takes for a population to reach a specific size. Students will apply their knowledge to find the time required for fish to reach a set population.
This module introduces power series solutions for differential equations. Students will learn how power series can simplify solving complex differential equations, making the process more approachable and manageable.
This module provides a detailed exploration of the analytic solution to the logistic differential equation. By understanding this solution, students will deepen their knowledge of how to handle logistic models in various applications.
This module introduces exact differential equations, explaining what it means for a differential equation to be exact. Students will learn how to identify and solve an exact differential equation through practical examples.
This module focuses on first-order linear differential equations, providing students with essential techniques and methods for solving these types of equations effectively. The concepts are fundamental for advanced studies in differential equations.