Differential Equations In Depth

Name: Differential Equations In Depth
Rating: 4.5 (233 reviews)

An in-depth course on differential equations, covering first/second order ODEs, PDEs and numerical methods, too!

Created byDmitri Nesteruk

Last updated 7/2017

English

What you'll 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

Course content

8 sections • 48 lectures • 3h 17m total length

Course Introduction0:58
Some information about the course instructor and the structure of the course.
A Word on Notation8:22
A note on the notation (Leibniz, Lagrange, Newton) that is used for the derivatives.
Chain Rule3:05
A look at the differentiation of compositions of functions.
Implicit Differentiation3:57
A look at implicitly defined functions and how to differentiate them.
Let's Make a Differential Equation!0:48
Before we learn how to solve differential equations, let's take a look at how to actually make one!
Equation Order0:44
One key characteristic of a differential equation is its order.
Verifying Solutions1:25
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.

Direct Integration1:48
A look at the direct integration approach, which helps us solve the simplest differential equations.
Separation of Variables3:42
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.
Homogeneous Functions3:56
An explanation of the concept of a homogeneous function of one or more variables. This will be useful for solving homogeneous ODEs.
Homogeneous Equations6:18
A look at homogeneous equations and how to solve them using the y=vx substitution.
Linear Equations4:32
A look at linear first-order differential equations and the use of the integrating factor.
Bernoulli's Equations9:22
A look at Bernoulli's equations, which are also first-order but non-linear due to an yⁿ term on the right.
Exact Equations8:06
A look at the method of exact equations.
Solving DIfferential Equations in MATLAB3:08
A look at MATLAB's support for symbolic differentiation, integration and solving differential equations.
Summary1:47
A summary of all the methods we've seen for solving 1st-order ODEs.

Reduction of Order5:28
A look at the Reduction of Order technique which helps us turn some higher-order (not necessarily linear) equations into first-order ones.
Second-Order Linear2:53
A look at Second-Order Linear equations with constant coefficients.
Two Distinct Real Roots1:15
A solution with an auxiliary equation that yields two real roots.
Single Real Root1:00
A solution with an auxiliary equation that yields a single real root.
Complex Roots1:28
A solution with an auxiliary equation that yields complex roots.

Introduction2:01
An introduction to the idea of transforms and, specifically, the Laplace transform.
A Few Transforms2:48
A look at a few Laplace transforms of simple functions.
Inverse Transform2:36
The inverse transform is just as useful as the transform itself!
Transform of a Derivative3:56
Laplace transform of first and second derivatives of a function.
Laplace Transforms in MATLAB4:58
A look at how to find to find direct and inverse Laplace transforms in MATLAB.
Differentiating Laplace Transforms5:56
A look at how to differentiate Laplace transforms.
Solving Differential Equations4:20
Finally, leveraging the Laplace transform to simplify solving differential equations.
Integration of a Laplace Transform5:38
Having differentiated Laplace transforms, how about integrating one?

Introduction1:49
Introducing Fourier Series.
Integrals of Periodic Functions4:09
A look at how to integrate periodic functions.
Orthogonality0:56
Introducing the concept of mutually orthogonal functions.
Fourier Series0:57
Finally, a look at the Fourier Series itself.
Fourier Coefficients3:01
An explanation of how to calculate the coefficients in the Fourier Series for a particular function f(x).

Partial Derivatives4:30
A look at what partial derivatives are.
Notation1:37
Ways of actually writing down partial derivatives.
Partial Differential Equations2:01
A look at the general form of a 2nd-order linear PDE.
Boundary Conditions12:27
We solve a pair of differential equations with boundary conditions.
Symbolic Solutions in Maple6:22
A look at how to solve PDEs symbolically - in Maple, not MATLAB.
Heat Conduction Equation4:29
Wave Equation13:27
An equation that describes the propagation of waves.
Wave Equation Example3:55
A practical example of finding an exact solution to the wave equation.

Introduction2:14
Introducing numerical methods.
Euler's Method8:51
Our first attempt at finding a numeric solution to differential equations.
Improved Euler's Method4:00
An attempt to improve upon Euler's method by treating the area under the curve as a trapezoid instead of a rectangle.
Runge-Kutta Method6:56
A significant improvement upon Euler's method, representing the slice between the curves as a parabola.
Numerical Approximation of Derivaties9:30
A look at forward, backward and central difference approximations of first and second derivatives of functions.

Requirements

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

Description

This course has everything you need to learn and understand Differential Equations. Differential equations are a class of equation that involves the use of differentials (derivatives) in their construction. Differential equations are used in many areas of science, particularly in physics, where they are used to model real-world phenomena such as the propagation of waves.

This course covers:

Ordinary differential equations (ODEs) - first and second order
Laplace Transform and Fourier Series
Partial differential equations (PDEs) - including common equations such as the Wave Equation and the Heat Conduction Equation
Numeric solutions of differential equations - e.g., Euler's Method, Runge-Kutta
Modeling and solving differential equations using MATLAB and Maple.

Course pre-requisites:

Fundamental understanging of differentiation and integration
Knowledge of common integration operations (integration by parts, integration by substitution)
Basic understanding of numerical computing (required for numerical methods)
Access and basic knowledge of common CAS packages such as MATLAB, Maple, Mathematica, etc.

This course is presented as a series of hand-written lectures where we discuss the relevant topics. I also present approaches to using CAS (Computer-aided Algebra Systems) to solving differential equations either analytically or symbolically.

It is recommended that you augment your study of differential equations on this course with a good textbook on differential equations.

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

Who this course is for:

College students
University students
Math enthusiasts

Differential Equations In Depth

What you'll learn

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Course content

Introduction to Differential Equations7 lectures • 19min

First-Order Differential Equations9 lectures • 43min

Second-Order Differential Equations5 lectures • 12min

Laplace Transform8 lectures • 32min

Fourier Series5 lectures • 11min

Partial Differential Equations8 lectures • 49min

Numerical Methods5 lectures • 32min

End of Course1 lecture • 1min

Requirements

Description

Who this course is for: