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Differential Equations In Depth

An in-depth course on differential equations, covering first/second order ODEs, PDEs and numerical methods, too!
4.0 (24 ratings)
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306 students enrolled
Last updated 8/2014
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  • 3.5 hours on-demand video
  • 1 Supplemental Resource
  • Full lifetime access
  • Access on mobile and TV
  • Certificate of Completion
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What Will I Learn?
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
View Curriculum
  • Good knowledge of calculus (linear algebra, differentiation, integration)

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.

Who is the target audience?
  • College students
  • University students
  • Math enthusiasts
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Curriculum For This Course
Expand All 47 Lectures Collapse All 47 Lectures 03:17:26
Introduction to Differential Equations
7 Lectures 19:19

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

Preview 00:58

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

Preview 08:22

A look at the differentiation of compositions of functions.

Chain Rule

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

Implicit Differentiation

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

Let's Make a Differential Equation!

One key characteristic of a differential equation is its order.

Equation Order

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.

Verifying Solutions
First-Order Differential Equations
9 Lectures 42:39

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

Direct Integration

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.

Separation of Variables

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

Preview 03:56

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

Homogeneous Equations

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

Linear Equations

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

Bernoulli's Equations

A look at the method of exact equations.

Exact Equations

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

Solving DIfferential Equations in MATLAB

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

Second-Order Differential Equations
5 Lectures 12:04

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

Reduction of Order

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

Second-Order Linear

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

Two Distinct Real Roots

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

Single Real Root

A solution with an auxiliary equation that yields complex roots.

Complex Roots
Laplace Transform
8 Lectures 32:13

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


A look at a few Laplace transforms of simple functions.

A Few Transforms

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

Inverse Transform

Laplace transform of first and second derivatives of a function.

Transform of a Derivative

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

Laplace Transforms in MATLAB

A look at how to differentiate Laplace transforms.

Differentiating Laplace Transforms

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

Solving Differential Equations

Having differentiated Laplace transforms, how about integrating one?

Integration of a Laplace Transform
Fourier Series
5 Lectures 10:52

Introducing Fourier Series.


A look at how to integrate periodic functions.

Integrals of Periodic Functions

Introducing the concept of mutually orthogonal functions.


Finally, a look at the Fourier Series itself.

Fourier Series

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

Fourier Coefficients
Partial Differential Equations
8 Lectures 48:48

A look at what partial derivatives are.

Partial Derivatives

Ways of actually writing down partial derivatives.


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

Partial Differential Equations

We solve a pair of differential equations with boundary conditions.

Boundary Conditions

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

Symbolic Solutions in Maple

Heat Conduction Equation

An equation that describes the propagation of waves.

Wave Equation

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

Wave Equation Example
Numerical Methods
5 Lectures 31:31

Introducing numerical methods.


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

Euler's Method

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

Improved Euler's Method

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

Runge-Kutta Method

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

Numerical Approximation of Derivaties
About the Instructor
4.5 Average rating
167 Reviews
1,182 Students
9 Courses
Quant Finance • Algotrading • Software/Hardware Engineering

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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