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30-Day Money-Back Guarantee

This course includes:

  • 29.5 hours on-demand video
  • 14 downloadable resources
  • Full lifetime access
  • Access on mobile and TV
Teaching & Academics Math Differential Equations

A Complete First Course in Differential Equations

A University Level Introductory Course in Differential Equations
Rating: 4.2 out of 54.2 (444 ratings)
4,062 students
Created by Chris Levy
Last updated 3/2016
English
English [Auto]
30-Day Money-Back Guarantee

What you'll learn

  • Classify differential equations according to their type and order.
  • Solve first order differential equations that are separable, linear, homogeneous, exact, as well as other types that can be solved through different substitutions.
  • Use first order differential equations to model different applications from science.
  • Solve linear second order equations with constant coefficients (both homogenous and non-homogeneous) using the method of undetermined coefficients, variation of parameters, and Laplace transforms.
  • Understand the theory of linear second order differential equations and how it relates to ideas from linear algebra.
  • Use linear second order equations with constant coefficients (both homogenous and non-homogeneous) to model applications from science.
  • Find Laplace and inverse Laplace transforms.
  • Use Laplace transforms to solve linear second order equations with constant coefficients which contain forcing functions such as impulses, step functions, and periodic functions.
  • Solve systems of linear differential equations with constant coefficients and understand the importance of eigenvalues and eigenvectors for finding solutions.
  • Understand the importance of the Matrix exponential and how to compute it in order to find the solutions of linear systems of differential equations.
  • Apply basic numerical methods to find approximate solutions of differential equations.
  • Understand the basics of some complex analysis and its usefulness to differential equations.
  • Use equilibrium points, phase portraits, and stability analysis to analyze linear systems.
  • Use Maple to analytically and numerically solve differential equations. Use Maple to study differential equations qualitatively.
  • Model real world phenomenon with differential equations.
  • Find series solutions to second order linear equations with variable coefficients. Apply this method to ordinary points and regular singular points. Find Frobenius series solutions using the method of Frobenius. Apply reduction of order to find series solutions.
  • Use Fourier series to solve partial differential equations. Solve the heat, wave, and Laplace equation using separation of variables and Fourier Series. Understand theory and applications of General Fourier series, Sine Fourier series, Cosine Fourier series, and convergence of Fourier series. Solve inhomogenous PDEs.
  • Use theory of vector spaces, orthogonality of functions and inner products, self adjoint operators and apply to Sturm-Liouville Eigenvalue problems. Use eigen function expansions to solve nonhomogenous problems.
  • Analyze nonlinear autonomous system by finding equilibrium points and stability. Understand concept of linearization and the Hartman-Grobman Theorem. Find and analyze Hopf bifurcation as well as other commonly known bifurcations
  • Apply Numerical methods and understand importance of stability and accuracy. Be able to implement in Maple. Be able to use state of the art DE solvers.

Requirements

  • First year differential and integral calculus

Description

This course will teach everything that is usually taught in the first two semesters of a university/college course in differential equations. The topics we will consider in this course are

  1. First Order Differential Equations
  2. Linear Equations of Higher Order
  3. Laplace Transform Methods
  4. Linear Systems of Differential Equations
  5. Power Series Methods
  6. Partial Differential Equations
  7. Fourier Series
  8. Sturm Liouville Eigenvalue Problems
  9. Nonlinear Systems of Differential Equations
  10. Numerical Methods

Who this course is for:

  • Students taking differential equations at college or university
  • Students preparing to take differential equations at college or university
  • Anyone who wants to learn about the subject of differential equations

Course content

9 sections • 173 lectures • 30h 42m total length

  • Preview06:55
  • Object falling under the force of gravity and air resistance
    04:11
  • Motion of a mass on a spring
    07:03
  • RLC Circuits
    07:23
  • Motion of a simple pendulum
    06:45
  • More Differential Equation Models
    12:45
  • Defining and Classifying Differential Equations
    13:30
  • Solutions of Differential Equations
    07:42
  • Explicit and Implicit Solutions
    04:18

  • Slope Fields and Solution Curves
    11:28
  • Existence and Uniqueness for first order Differential Equations
    10:45
  • Separable Differential Equations
    06:14
  • Separable Differential Equation Examples
    05:39
  • Newtons Law of Cooling
    07:05
  • Newtons Law of Cooling: Homicide Victim Example
    05:57
  • Torricellis Law
    10:16
  • Torricellis Law Example
    13:02
  • Linear First Order Differential Equations
    14:52
  • Differential Equation for Mixing Problems
    07:47
  • Mixing Problem Example
    09:17
  • Exact Differential Equations
    07:22
  • Exact Differential Equation Example 1
    10:23
  • Exact Differential Equation Example 2
    11:54
  • Preview06:48
  • Homogenous Differential Equations
    09:09
  • Homogeneous Differential Equation Example 1
    08:58
  • Preview03:26
  • Bernoulli Differential Equations
    08:04
  • Preview10:56
  • Preview14 pages

  • Higher Order Differential Equations
    05:41
  • Linear Differential Operators
    04:28
  • Principal of Superposition
    07:01
  • Existence and Uniqueness Theorem
    04:17
  • The Wronskian Determinant
    11:44
  • General Solutions of Second Order Linear Homogenous Equations
    09:20
  • Summary of Theory for Second Order Homogenous Equations
    02:01
  • Linear Independence and the Wronskian
    12:02
  • Wronskian of Solutions
    05:32
  • Theory of Higher Order Equations
    07:48
  • Solving Second Order Equations with Constant Coefficients
    03:56
  • Second Order Equations with Constant Coefficients: Distinct Roots
    04:26
  • Solving Second Order Equations with Constant Coefficients: 1 Root
    04:49
  • Solving Second Order Equations with Constant Coefficients: Complex Roots
    11:30
  • Method of Reduction
    12:30
  • Higher Order Equations: Distinct Real Roots
    03:24
  • Higher Order Equations: Repeated Real Roots
    03:57
  • Higher Order Equations: Distinct Complex Roots
    02:13
  • Higher Order Equations: Repeated Complex Roots
    03:37
  • Higher Order Equations: Example With All Cases
    02:57
  • Nonhomogenous Differential Equations
    03:18
  • Method of Undetermined Coefficients Example 1
    08:48
  • Method of Undetermined Coefficients Example 2
    03:48
  • Method of Undetermined Coefficients Example 3
    07:05
  • Method of Undetermined Coefficients: Avoiding Duplication
    08:32
  • Method of Undetermined Coefficients In General
    03:32
  • Method of Undetermined Coefficients Example 4
    07:50
  • Method of Undetermined Coefficients Example 5
    06:23
  • Method of Undetermined Coefficients Example 6
    04:10
  • Assignment 2
    4 pages
  • Reduction of Order: The General Formula
    09:45
  • Reduction of Order: An Example
    04:11
  • Variation of Parameters
    15:32
  • Variation of Parameters: An Example
    05:20
  • Assignment 3
    4 pages

  • The Laplace Transform
    05:17
  • Laplace Transform Example: Unit Step Function
    04:50
  • Laplace Transform Example: First Derivative
    04:27
  • Laplace Transform Example: Second Derivative
    03:10
  • Existence of the Laplace Transform
    08:08
  • Laplace Transform Example: Exponential Function
    02:27
  • Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
    08:18
  • The Inverse Laplace Transform
    03:58
  • Solving Differential Equations with Laplace Transform
    11:10
  • Solving Differential Equations with Laplace Transform
    08:30
  • Partial Fractions to Invert Transforms
    12:35
  • First Translation Theorem
    05:46
  • First Translation Theorem: Inverting Transforms
    08:35
  • First Translation Theorem: Inverting Transforms: Completing Square
    07:59
  • Second Translation Theorem
    03:23
  • Piecewise Continuous Functions with Unit Step Functiond
    06:59
  • Laplace Transform of Piecewise Continuous Functions
    04:16
  • Laplace Transform of Piecewise Continuous Functions
    05:29
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
    19:45
  • Solving an IVP with a Piecewise continuous Non-homogenous Term
    13:12
  • Assignment 4
    5 pages
  • Derivatives of Transforms
    07:42
  • Laplace Transform of Piecewise Periodic Functions
    05:30
  • Solving an IVP with a Piecewise Periodic Non-homogenous Term
    16:42
  • The Dirac Delta Function
    10:55
  • Solving an IVP with a Delta Function Term
    09:10
  • Solving an IVP with Multiple Delta Function Term
    05:19
  • The Convolution Theorem
    06:58
  • Convolution Theorem: Finding Integral Solutions
    04:18
  • Convolution Theorem: Finding Integral Solutions
    06:31
  • Assignment 5
    7 pages

  • Power Series Template Slides
    12 pages
  • Review of Second Order Equations (Constant Coefficients) and Power Series
    14:24
  • Solving Airy's Differential Equation with Power Series Solution
    22:02
  • Plotting Solutions of Airy's DE and using Maple to find Series Solutions
    19:49
  • Finding a Power Series Solution, Using Maple as well, Ordinary Points
    19:08
  • Ordinary Points. Chebyshev's Differential Equation
    20:13
  • Previous Video Continued: Chebyshev Polynomials
    05:51
  • Quiz: Power Series Solution about Ordinary Point
    1 page
  • Quiz Solution: Power Series Solution about Ordinary Point
    2 pages
  • Singular Points. Regular Singular Points. Euler's Differential Equation
    15:12
  • Euler's Differential Equation Continued
    11:33
  • Frobenius Series Solutions and Beginning of Example
    13:15
  • Frobenius Series Solution: Roots Differing by Non Integer
    15:17
  • Frobenius Series: Roots Differing by an Integer - 2 Frobenius Solutions
    04:58
  • Roots Differing by an Integer - 1 Frobenius Solution Continued
    18:39
  • Frobenius Series: Roots Differing by an Integer- 1 Frobenius Solution
    11:30
  • Quiz: Frobenius Series
    1 page
  • Method of Reduction with Frobenius Series
    06:53
  • Method of Reduction with Frobenius Series Continued
    22:22
  • Quiz: Frobenius and Reduction of Order
    1 page

  • Partial Differential Equations and Fourier Series Template Slides
    14 pages
  • Intro to PDEs
    21:38
  • Separation of Variables: Heat Equation - Zero B.C.
    18:58
  • Heat Equation - Zero B.C. - Sine Fourier Series
    18:45
  • Heat Equation - Zero B.C. - Sine Fourier Series Continued
    22:52
  • Sine Fourier Series Continued and Heat Equation
    10:29
  • Heat Equation - Zero Flux B.C. - Cosine Fourier Series
    25:46
  • Heat Equation - Periodic B.C. General Fourier Series
    18:23
  • Fourier Series Continued
    17:01
  • Quiz - Fourier Series
    1 page
  • Fourier Series of Piecewise Continuous Function
    23:27
  • Fourier Series - Convergence
    11:42
  • Cosine and Sine Fourier Series - Even and Odd Extensions
    17:33
  • Removing Inhomogeneous Terms in PDE: Heat Equation
    22:36
  • Removing Inhomogeneous Terns in Boundary Conditions: Heat Equation
    07:45
  • Wave Equation
    22:47

  • Slides for This Chapter
    16 pages
  • Self Adjoint Operators
    23:28
  • Regular Sturm-Liouville Eigenvalue Problems
    12:53
  • Regular Sturm-Liouville Operator is Self Adjoint
    18:36
  • Regular Sturm-Liouville: Orthogonal Eigen Functions and Real Eigenvalues
    15:08
  • Regular Sturm-Liouville Theorem and Eigen Function Expansions
    12:12
  • Converting DEs to Sturm Liouville FOrm
    12:26
  • Sturm Liouville Example with Euler's Equation
    22:28
  • Nonhomogenous Sturm Liouville Problem
    20:39

  • Intro. Ex. of First Order Nonlinear DEs : Equilibriums, Stability, Maple!
    18:24
  • Intro. Ex. of First Order DEs : Equilibriums, Stability, Maple! Continued...
    16:19
  • Logistic growth with constant harvest, equilib. points, stability, bifurcation
    17:43
  • Logistic growth with constant harvest, numerical solve Maple
    04:15
  • Logistic periodic harvesting, equilibrium and stability defn. for systems
    16:07
  • Review of Linear Systems Phase Portraits
    18:23
  • Linearization of Nonlinear System - Jacobian - Example as well
    22:11
  • Equilibrium Points, Stability, Phase Portrait, Numerical Solution in Maple
    15:13
  • Equilibrium Points, Stability, Phase Portrait, Numerical Solution in Maple
    17:19
  • Hopf Bifurcation Part 2
    19:43
  • Hopf Bifurcation Part 2
    18:59

  • Intro to Euler's Method
    17:12
  • Euler's Method Example (by hand)
    13:11
  • Euler's Method in Maple
    07:03
  • Euler's Method in Excel
    06:06
  • Euler's Method In Maple (Another Example) and Dsolve Numeric in Maple
    10:00
  • Stability of Euler's Method Part 1
    07:00
  • Stability of Euler's Method Part 2
    13:14
  • Backward/Implicit Euler Part 1
    24:46
  • Implicit Trapezoid Stability
    19:07
  • Accuracy of Euler Method
    13:15
  • Accuracy of Implicit Trapezoid Method
    09:15
  • Hints for an assignment #7. Random Review Stuff. For Students in Math 3120
    17:29
  • Coding Implicit Trapezoid
    10:04
  • Heun's Method
    04:09
  • Runge Kutta (RK2) Derivation
    20:17
  • Runge Kutta (RK4) Method
    17:26
  • Review of some Numerical Stability Concepts and the Methods we have looked at
    08:02
  • Polynomial Interpolation (Vandermonde Matrix)
    11:41
  • Newton Divided Difference Polynomial Interpolation
    17:30
  • Backward Differentiation BDF1
    14:31
  • Backward Differentiation BDF2
    13:01

Instructor

Chris Levy
Research Scientist / Data Scientist / PhD Applied Math
Chris Levy
  • 4.4 Instructor Rating
  • 1,742 Reviews
  • 22,280 Students
  • 5 Courses

I left the world of academia for a career in data science. Through  years of industry experience I have developed skills as a full stack data scientist in the areas of data engineering, machine learning, data visualization, and productizing data science models. More recently I have been working as a research scientist with a focus on computer vision and deep learning.

I  have experience teaching thousands of students in mathematics at the university level as well as thousands of students on Udemy in areas like math, SQL, and coding. I  love to learn and teach others what I am learning.

I enjoy hanging out with my wife and three kids, playing guitar, playing basketball, biking, and learning.

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