Here, you find my whole video series about Ordinary Differential Equations in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:
Part 1 - Introduction
Ordinary Differential Equations is a video series I started for everyone who is interested in the theory of differential equations and dynamics and also in concrete solution methods.:
With this you now know the topics that we will discuss in this series. Now, in the next video let us discuss definitions in the topic.
Part 2 - Definitions
There are a lot of notions one has to define to get a precise definition what a differential equation is and what it means to find solutions for it. Let’s do it:
Part 3 - Directional Field
After defining all these new manners of speaking, we are ready to go to a more visual approach. The direction field is essentially just a graph visualisation of a vector-valued function but it helps a lot to get properties from a given differential equation. We will first look at one- and two-dimensional examples.
Part 4 - Reducing to First Order
We already used it in the last video: the fact that every ordinary differential equations could be described as a system of first order. Now we will fill in these gaps and show the correct substitutions, one needs to get this result.
Part 5 - Solve First-Order Autonomous Equations
In this video, we will consider a general one-dimensional differential equation of first order. It turns out that we can describe the solution just as an antiderivate.
Part 6 - Separation of Variables
Now, we are ready to generalise the result from the last video. We can do that in a special situation when two variables can be separated.
Part 7 - Solving Linear Equations of First Order
In this video, we learn about the method of integrating factor to solve linear ODEs. This is equivalent to the method of variation of the constant, which is often taught in this context.
Part 8 - Existence and Uniqueness?
In this video, we start discussing if solutions of initial value problems can exist and if we even find uniquness. By looking at the directional field, we conclude that existence should be given but uniqueness is not garantueed. However, we will give sufficient conditions for unique solutions in future videos.
Part 9 - Lipschitz Continuity
Part 10 - Uniqueness for Solutions
Now we are ready to prove the uniqueness of a solution for a given initial value problem if a Lipschitz condition is satisfied.