A Complete First Course in Differential Equations

A University Level Introductory Course in Differential Equations
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  • Lectures 173
  • Length 30.5 hours
  • Skill Level All Levels
  • Languages English
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About This Course

Published 9/2015 English

Course 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

What are the requirements?

  • First year differential and integral calculus

What am I going to get from this course?

  • 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.

What is the target audience?

  • 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

What you get with this course?

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Curriculum

Section 1: Introduction to Differential Equations and their Applications
06:55

Here we look at a differential equation which models the motion of a falling object under the force of gravity.

04:11

Here we look at a differential equation which models the motion of a falling object under the force of gravity and air resistance.

07:03

Here we look at a differential equation which models the motion of mass on a spring.

07:23

Here we look at a differential equation which models RLC circuits. It is very interesting to find that the governing differential equation has the exact same form as the differential equation which models the motion of a mass on a spring.

06:45

Here we look at a differential equation which models the motion of a simple pendulum.

12:45

Here we look at several differential equations which occur in applications. We consider the equation for a hanging rope, Newtons law of cooling, the deflection of a cantilever beam, and a simple population growth model.

13:30

In this video we define what an ordinary differential equation is and also how to classify them in terms of their order and whether they are linear or nonlinear.

07:42

Here we define what it means for a differential equation to have a solution. We do some simple examples where we verify a function is a solution to a differential equation.

04:18

Here we show that solutions to differential equations can be explicit or implicit.

Section 2: First Order Differential Equations
11:28

In this video we learn about slope fields and solution curves for first order differential equations.

10:45

In this video we learn about the existence and uniqueness theorem for first order differential equations.

06:14

In this video we define what separable differential equations are and learn how to solve them.

05:39

In this video we solve a separable differential equation. We also quickly review integration by parts.

07:05

In this video we discuss the differential equation for Newtons law of cooling and solve it. We can solve the equation because its separable.

05:57

In this video we learn how to find the time of death when a homicide victim is found in a room with some temperature. This is an application of Newtons law of cooling.

10:16

In this video we use Torricellis Law and calculus to derive the differential equation for water draining from a tank with a hole in its base.

13:02

Here we solve a differential equation for the height of water draining out of a conical water tank. Using our solution we can calculate how long it takes for the tank to drain.

14:52

In this video we learn to solve linear first order differential equations.

07:47

Here we derive a differential equation for simple mixing problems. Mixing problems are a nice application of first order linear differential equations. In these problems a solute/solvent mixture is added to a tank with a similar mixture. The mixture is then pumped out of the tank. A differential equation for the amount of solute in the tank is derived.

09:17

In this video we do an example of a mixing problem with one tank.

07:22

In this video we learn how to solve exact differential equations.

10:23

In this video we go through an example of solving an exact differential equation.

11:54

In this video we go through another example of solving an exact differential equation.

06:48

In this video we begin to look at substitution methods. In particular we learn how to solve differential equations of the form dy/dx=f(ax+by+c) by making an appropriate substitution.

09:09

In this video we define what first order homogenous differential equations are. We also define homogeneous functions. These homogeneous differential equations can be solved by a substitution. This video shows how this is done.

08:58

In this video we go through an example of solving a first order homogeneous differential equation.

03:26

In this video we go through a second example of solving a first order homogeneous differential equation.

08:04

In this video we learn how to solve Bernoulli Differential Equations. They can be solved by an appropriate substitution.

10:56

In this video we see how substitutions can sometimes reduce second order differential equations to first order differential equations which we can then solve.

14 pages

This assignment tests your understanding of the material presented in Sections 1 and 2.

Section 3: Higher Order Differential Equations
05:41

In this video we begin to look at higher order differential equations. In particular we focus on the theory of linear seconder order differential equations. We define a differential operator that we will be using through the next several videos. We also define homogeneous second order equations and non-homogeneous equations.

04:28

In this video we define what a linear operator is and discuss how the 2nd order differential operator defined in the previous video is a linear operator.

07:01

In this video we look at an important theorem known as the principal of superposition for linear homogeneous equations. It says that if y1 and y2 are two solutions to a homogeneous equation, then so is any linear combination of them.

04:17

In this video we state the existence and uniqueness theorem for higher order differential equations.

11:44

In this video we define the Wronksian determinant and see its importance in the theory of linear differential equations. In particular, if y1 and y2 are two different solutions to a second order linear homogeneous DE, then constants c1 and c2 can be chosen to solve an IVP provided that the Wronskian of y1 and y2 is non zero.

09:20

This video continues from the previous one. We state a main theorem summarizing the information from the last video while also adding another important statement. We discuss how every solution to linear homogeneous differential equations can be written in a certain way.

02:01

In this video we summarize the main points of interest regarding the solutions of linear second order homogeneous equations.

12:02

Here we introduce the idea of linear independence and dependence for two functions and relate it to the Wronskian of two functions.

05:32

In this video we look at a stronger theorem relating linear independence/dependence to the wronskian for solutions of second order linear homogenous equations.

07:48

In this video we discuss the theory of nth order linear homogeneous equations. We learn about linear independence/dependence and the wronskian for n>2 functions. We also discuss whats needed for general solutions. The theory is just an extension of the theory for 2nd order equations we have seen in the last several videos.

03:56

In this video we begin to learn how to solve second order homogeneous equations with constant coefficients.

04:26

In this video we learn how to solve second order homogeneous equations with constant coefficients when the roots of the characteristic equation are real and distinct.

04:49

In this video we learn how to solve second order homogeneous equations with constant coefficients when there is only one real root of the characteristic equation.

11:30

In this video we learn how to solve second order homogeneous equations with constant coefficients when there is complex roots of the characteristic equation.

12:30

In this video we discuss how we can find a second solution to a second order equation when we already know one solution. We do this in the specific case of having one root of the characteristic equation.

03:24

In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are distinct real roots.

03:57
In this video we learn how to solve higher order linear constant coefficient equations when the some of the roots of the characteristic equation are repeated real roots.
02:13
In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are complex (but not repeated).
03:37
In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are complex and repeated.
02:57
In this video we look at solving one more higher order homogeneous equation with constant coefficients. We see how the different cases from the last few videos can be used together in one differential equation.
03:18
In this video we begin to discuss the solutions of nonhomogenous differential equations.
08:48

In this video we solve our first nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a polynomial.

03:48
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is an exponential function.
07:05
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a cosine function.
08:32
In this video we consider another nonhomogenous differential equation and try using the method of undetermined coefficients as we have done in the last few videos. In turns out that we have to modify our guess of the particular solution when terms in it also appear in the complementary solution.
03:32

In this video we consider the method of undetermined coefficients in general. We explain when it works and how to come up with a particular solution based off the right hand side of the differential equation (the nonhomogenous term) and the complementary solution.

07:50
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is an exponential function multiplied by a cosine function.
06:23
In this video we consider another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a sum of two nonhomogenous terms. For each term we find particular solutions which then need to be added together to find the general particular solution. This video also shows what to do when there is duplication between the particular solution and the complementary solution.
04:10

In this video we consider another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a sum of two nonhomogenous terms. For each term we find particular solutions which then need to be added together to find the general particular solution. This video also shows what to do when there is duplication between the particular solution and the complementary solution.

Assignment 2
4 pages
09:45
If solving a linear second order homogenous differential equation, we can always find a second solution if we already know one solution. This is done through the method of reduction of order. We had considered this method for a specific example in an earlier video. Here we derive a formula in the general case which can be used to find the second solution.
04:11
In this video we use the reduction of order formula to find a second solution for a specific example.
15:32
In this video we derive a formula for finding a particular solution to a nonhomogenous differential equation. This method is an alternative method to the method of undetermined coefficients.
05:20
In this video we use the method of variation of parameters to solve a nonhomogenous differential equation.
Assignment 3
4 pages
Section 4: Laplace Transforms
05:17
In this video we define the Laplace transform and discuss the types of differential equations we will be solving with it.
Laplace Transform Example: Unit Step Function
04:50
Laplace Transform Example: First Derivative
04:27
03:10
In this video we find the Laplace transform of the second derivative of a function.
Existence of the Laplace Transform
08:08
02:27
In this video we find the Laplace transform of the exponential function.
Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine
08:18
03:58
In this video we define the inverse Laplace transform.
11:10

In this video we use Laplace transforms to solve our first initial value problem (differential equation with initial conditions).

08:30
In this video we use Laplace transforms to solve our second initial value problem (differential equation with initial conditions).
12:35
In this video we look at some more techniques to invert Laplace transforms. We use partial fractions along with a table of Laplace transforms.
05:46
Here we learn a very important theorem, the First Translation Theorem. It is useful in finding inverse Laplace transforms that are translated/shifted.
08:35
In this video we use the first translation theorem to find an inverse Laplace transform.
07:59

In this video we use the first translation theorem to find an inverse Laplace transform.

Completing the square is an important concept here.

03:23
In this video we learn about another very important translation theorem which we will call the Second Translation Theorem. It involves finding the transforms of unit step functions which are multiplied by translated functions.
06:59

In this video we learn how to rewrite piecewise continuous functions in terms of the unit step function.

04:16

In this video we take the Laplace transform of a piecewise continuous step function.

05:29
In this video we take the Laplace transform of a piecewise continuous step function.
19:45
In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.
13:12
In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.
Assignment 4
5 pages
07:42
In this video we look at a formula for the derivative of transforms and one application of it.
05:30
In this video we find out how to take the Laplace Transform of periodic piecewise functions.
16:42
In this video we solve an IVP with a periodic piecewise nonhomogenous term.
10:55
In this video we define the Dirac Delta Function which can be used to model impulses.
09:10
In this video we solve an IVP with a delta function term.
05:19

In this video we solve an IVP with a series of delta function terms.

06:58

In this video we define the convolution of two functions and show the convolution theorem.

We also use it to find the inverse Laplace transform of a function.

04:18

In this video we use the convolution theorem to find a nice closed integral formula for the solution of a differential equation.

06:31

In this video we use the convolution theorem to find a nice closed integral formula for the solution of a differential equation. In particular we discuss the response of a system to a delta function input. Then we use the convolution theorem to find a nice closed integral formula for the solution of general second order linear differential equation. This formula has many nice applications and can be derived very easily.

Assignment 5
7 pages

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Instructor Biography

Chris Levy, PhD Applied Mathematics

My name is Chris Levy and I have a PhD in applied mathematics from Dalhousie University. I live in Halifax, Nova Scotia, Canada.

I am a researcher, university instructor, and a budding data scientist. I have experience teaching university courses such as calculus, differential equations, and math for commerce. I have taught courses to 35 students, 70 students, and even 300 students.

I also have experience tutoring hundreds of students in mathematics. I know how to explain concepts clearly and concisely. I have been a very successful student and instructor. I know what it takes to succeed.

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

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