Differential Equations In Depth

An in-depth course on differential equations, covering first/second order ODEs, PDEs and numerical methods, too!
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  • Lectures 47
  • Contents Video: 3.5 hours
  • Skill Level All Levels
  • Languages English
  • Includes Lifetime access
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About This Course

Published 8/2014 English

Course Description

This course has everything you need to learn and understand Differential Equations. This course covers:

  • Ordinary differential equations (ODEs)
  • Laplace Transform and Fourier Series
  • Partial differential equations (PDEs)
  • Numeric solutions of differential equations
  • Modeling and solving differential equations using MATLAB

This course will continue to evolve and improve based on feedback from the course participants.

What are the requirements?

  • Good knowledge of calculus (linear algebra, differentiation, integration)

What am I going to get from this course?

  • Learn how to solve different types of differential equations
  • Discover tricks and shortcuts to find solutions quicker
  • Find out how to solve equations numerically as well as analytically
  • Learn to use MATLAB to solve differential equations

What is the target audience?

  • College students
  • University students
  • Math enthusiasts

What you get with this course?

Not for you? No problem.
30 day money back guarantee.

Forever yours.
Lifetime access.

Learn on the go.
Desktop, iOS and Android.

Get rewarded.
Certificate of completion.

Curriculum

Section 1: Introduction to Differential Equations
00:58

Some information about the course instructor and the structure of the course.

08:22

A note on the notation (Leibniz, Lagrange, Newton) that is used for the derivatives.

03:05

A look at the differentiation of compositions of functions.

03:57

A look at implicitly defined functions and how to differentiate them.

00:48

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

00:44

One key characteristic of a differential equation is its order.

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: First-Order Differential Equations
01:48

A look at the direct integration approach, which helps us solve the simplest differential equations.

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.

03:56

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

06:18

A look at homogeneous equations and how to solve them using the y=vx substitution.

04:32

A look at linear first-order differential equations and the use of the integrating factor.

09:22

A look at Bernoulli's equations, which are also first-order but non-linear due to an yn term on the right.

08:06

A look at the method of exact equations.

03:08

A look at MATLAB's support for symbolic differentiation, integration and solving differential equations.

01:47

A summary of all the methods we've seen for solving 1st-order ODEs.

Section 3: Second-Order Differential Equations
05:28

A look at the Reduction of Order technique which helps us turn some higher-order (not necessarily linear) equations into first-order ones.

02:53

A look at Second-Order Linear equations with constant coefficients.

01:15

A solution with an auxiliary equation that yields two real roots.

01:00

A solution with an auxiliary equation that yields a single real root.

01:28

A solution with an auxiliary equation that yields complex roots.

Section 4: Laplace Transform
02:01

An introduction to the idea of transforms and, specifically, the Laplace transform.

02:48

A look at a few Laplace transforms of simple functions.

02:36

The inverse transform is just as useful as the transform itself!

03:56

Laplace transform of first and second derivatives of a function.

04:58

A look at how to find to find direct and inverse Laplace transforms in MATLAB.

05:56

A look at how to differentiate Laplace transforms.

04:20

Finally, leveraging the Laplace transform to simplify solving differential equations.

05:38

Having differentiated Laplace transforms, how about integrating one?

Section 5: Fourier Series
01:49

Introducing Fourier Series.

04:09

A look at how to integrate periodic functions.

00:56

Introducing the concept of mutually orthogonal functions.

00:57

Finally, a look at the Fourier Series itself.

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
04:30

A look at what partial derivatives are.

01:37

Ways of actually writing down partial derivatives.

02:01

A look at the general form of a 2nd-order linear PDE.

12:27

We solve a pair of differential equations with boundary conditions.

06:22

A look at how to solve PDEs symbolically - in Maple, not MATLAB.

Heat Conduction Equation
04:29
13:27

An equation that describes the propagation of waves.

03:55

A practical example of finding an exact solution to the wave equation.

Section 7: Numerical Methods
02:14

Introducing numerical methods.

08:51

Our first attempt at finding a numeric solution to differential equations.

04:00

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

06:56

A significant improvement upon Euler's method, representing the slice between the curves as a parabola.

09:30

A look at forward, backward and central difference approximations of first and second derivatives of functions.

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Instructor Biography

Dmitri Nesteruk, Quantitative Finance Professional

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 high-performance 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 intro-level Quantitative Finance course, and has also made online video courses on CUDA, MATLAB, D, the Boost libraries and other topics.

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