Differential Equations In Depth
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Section 1: Introduction to Differential Equations  

Lecture 1  00:58  
Some information about the course instructor and the structure of the course. 

Lecture 2  08:22  
A note on the notation (Leibniz, Lagrange, Newton) that is used for the derivatives. 

Lecture 3  03:05  
A look at the differentiation of compositions of functions. 

Lecture 4  03:57  
A look at implicitly defined functions and how to differentiate them. 

Lecture 5  00:48  
Before we learn how to solve differential equations, let's take a look at how to actually make one! 

Lecture 6  00:44  
One key characteristic of a differential equation is its order. 

Lecture 7  01:25  
Once you have a solution to a differential equation, verifying a solution is easy: simply substitute the function, together with all necessary derivatives, back into the equation and see if the equality holds. 

Section 2: FirstOrder Differential Equations  
Lecture 8  01:48  
A look at the direct integration approach, which helps us solve the simplest differential equations. 

Lecture 9  03:42  
A look at an approach called separation of variables that helps us separate out the x and y terms to different sides of the equation for subsequent integration. 

Lecture 10  03:56  
An explanation of the concept of a homogeneous function of one or more variables. This will be useful for solving homogeneous ODEs. 

Lecture 11  06:18  
A look at homogeneous equations and how to solve them using the y=vx substitution. 

Lecture 12  04:32  
A look at linear firstorder differential equations and the use of the integrating factor. 

Lecture 13  09:22  
A look at Bernoulli's equations, which are also firstorder but nonlinear due to an y^{n} term on the right. 

Lecture 14  08:06  
A look at the method of exact equations. 

Lecture 15  03:08  
A look at MATLAB's support for symbolic differentiation, integration and solving differential equations. 

Lecture 16  01:47  
A summary of all the methods we've seen for solving 1storder ODEs. 

Section 3: SecondOrder Differential Equations  
Lecture 17  05:28  
A look at the Reduction of Order technique which helps us turn some higherorder (not necessarily linear) equations into firstorder ones. 

Lecture 18  02:53  
A look at SecondOrder Linear equations with constant coefficients. 

Lecture 19  01:15  
A solution with an auxiliary equation that yields two real roots. 

Lecture 20  01:00  
A solution with an auxiliary equation that yields a single real root. 

Lecture 21  01:28  
A solution with an auxiliary equation that yields complex roots. 

Section 4: Laplace Transform  
Lecture 22  02:01  
An introduction to the idea of transforms and, specifically, the Laplace transform. 

Lecture 23  02:48  
A look at a few Laplace transforms of simple functions. 

Lecture 24  02:36  
The inverse transform is just as useful as the transform itself! 

Lecture 25  03:56  
Laplace transform of first and second derivatives of a function. 

Lecture 26  04:58  
A look at how to find to find direct and inverse Laplace transforms in MATLAB. 

Lecture 27  05:56  
A look at how to differentiate Laplace transforms. 

Lecture 28  04:20  
Finally, leveraging the Laplace transform to simplify solving differential equations. 

Lecture 29  05:38  
Having differentiated Laplace transforms, how about integrating one? 

Section 5: Fourier Series  
Lecture 30  01:49  
Introducing Fourier Series. 

Lecture 31  04:09  
A look at how to integrate periodic functions. 

Lecture 32  00:56  
Introducing the concept of mutually orthogonal functions. 

Lecture 33  00:57  
Finally, a look at the Fourier Series itself. 

Lecture 34  03:01  
An explanation of how to calculate the coefficients in the Fourier Series for a particular function f(x). 

Section 6: Partial Differential Equations  
Lecture 35  04:30  
A look at what partial derivatives are. 

Lecture 36  01:37  
Ways of actually writing down partial derivatives. 

Lecture 37  02:01  
A look at the general form of a 2ndorder linear PDE. 

Lecture 38  12:27  
We solve a pair of differential equations with boundary conditions. 

Lecture 39  06:22  
A look at how to solve PDEs symbolically  in Maple, not MATLAB. 

Lecture 40 
Heat Conduction Equation

04:29  
Lecture 41  13:27  
An equation that describes the propagation of waves. 

Lecture 42  03:55  
A practical example of finding an exact solution to the wave equation. 

Section 7: Numerical Methods  
Lecture 43  02:14  
Introducing numerical methods. 

Lecture 44  08:51  
Our first attempt at finding a numeric solution to differential equations. 

Lecture 45  04:00  
An attempt to improve upon Euler's method by treating the area under the curve as a trapezoid instead of a rectangle. 

Lecture 46  06:56  
A significant improvement upon Euler's method, representing the slice between the curves as a parabola. 

Lecture 47  09:30  
A look at forward, backward and central difference approximations of first and second derivatives of functions. 
Dmitri Nesteruk is a developer, speaker and podcaster. His interests lie in software development and integration practices in the areas of computation, quantitative finance and algorithmic trading. His technological interests include C#, F# and C++ programming as well highperformance computing using technologies such as CUDA. He has been a C# MVP since 2009.
Dmitri is a graduate of University of Southampton (B.Sc. Computer Science) where he currently holds a position as a Visiting Researcher. He is also an instructor on an online introlevel Quantitative Finance course, and has also made online video courses on CUDA, MATLAB, D, the Boost libraries and other topics.