A Complete First Course in Differential Equations

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A University Level Introductory Course in Differential Equations

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1,208 students enrolled

What Will I 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

- First Order Differential Equations
- Linear Equations of Higher Order
- Laplace Transform Methods
- Linear Systems of Differential Equations
- Power Series Methods
- Partial Differential Equations
- Fourier Series
- Sturm Liouville Eigenvalue Problems
- Nonlinear Systems of Differential Equations
- Numerical Methods

Who 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

Compare to Other Differential Equations Courses

Curriculum For This Course

173 Lectures

30:42:13
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Introduction to Differential Equations and their Applications
9 Lectures
01:10:32

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

Preview
06:55

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

Object falling under the force of gravity and air resistance

04:11

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

Motion of a mass on a spring

07:03

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.

RLC Circuits

07:23

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

Motion of a simple pendulum

06:45

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.

More Differential Equation Models

12:45

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.

Defining and Classifying Differential Equations

13:30

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.

Solutions of Differential Equations

07:42

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

Explicit and Implicit Solutions

04:18

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First Order Differential Equations
21 Lectures
02:59:22

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

Slope Fields and Solution Curves

11:28

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

Existence and Uniqueness for first order Differential Equations

10:45

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

Separable Differential Equations

06:14

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

Separable Differential Equation Examples

05:39

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

Newtons Law of Cooling

07:05

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.

Newtons Law of Cooling: Homicide Victim Example

05:57

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.

Torricellis Law

10:16

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.

Torricellis Law Example

13:02

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

Linear First Order Differential Equations

14:52

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.

Differential Equation for Mixing Problems

07:47

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

Mixing Problem Example

09:17

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

Exact Differential Equations

07:22

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

Exact Differential Equation Example 1

10:23

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

Exact Differential Equation Example 2

11:54

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.

Preview
06:48

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.

Homogenous Differential Equations

09:09

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

Homogeneous Differential Equation Example 1

08:58

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

Preview
03:26

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

Bernoulli Differential Equations

08:04

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

Preview
10:56

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

Preview
14 pages

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Higher Order Differential Equations
35 Lectures
03:31:27

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.

Higher Order Differential Equations

05:41

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.

Linear Differential Operators

04:28

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.

Principal of Superposition

07:01

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

Existence and Uniqueness Theorem

04:17

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.

The Wronskian Determinant

11:44

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.

General Solutions of Second Order Linear Homogenous Equations

09:20

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

Summary of Theory for Second Order Homogenous Equations

02:01

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

Linear Independence and the Wronskian

12:02

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

Wronskian of Solutions

05:32

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.

Theory of Higher Order Equations

07:48

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

Solving Second Order Equations with Constant Coefficients

03:56

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.

Second Order Equations with Constant Coefficients: Distinct Roots

04:26

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.

Solving Second Order Equations with Constant Coefficients: 1 Root

04:49

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

Solving Second Order Equations with Constant Coefficients: Complex Roots

11:30

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.

Method of Reduction

12:30

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.

Higher Order Equations: Distinct Real Roots

03:24

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.

Higher Order Equations: Repeated Real Roots

03:57

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).

Higher Order Equations: Distinct Complex 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 and repeated.

Higher Order Equations: Repeated Complex Roots

03:37

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.

Higher Order Equations: Example With All Cases

02:57

In this video we begin to discuss the solutions of nonhomogenous differential equations.

Nonhomogenous Differential Equations

03:18

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.

Method of Undetermined Coefficients Example 1

08:48

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.

Method of Undetermined Coefficients Example 2

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 a cosine function.

Method of Undetermined Coefficients Example 3

07:05

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.

Method of Undetermined Coefficients: Avoiding Duplication

08:32

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.

Method of Undetermined Coefficients In General

03:32

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.

Method of Undetermined Coefficients Example 4

07:50

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.

Method of Undetermined Coefficients Example 5

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.

Method of Undetermined Coefficients Example 6

04:10

Assignment 2

4 pages

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.

Reduction of Order: The General Formula

09:45

In this video we use the reduction of order formula to find a second solution for a specific example.

Reduction of Order: An Example

04:11

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.

Variation of Parameters

15:32

In this video we use the method of variation of parameters to solve a nonhomogenous differential equation.

Variation of Parameters: An Example

05:20

Assignment 3

4 pages

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Laplace Transforms
31 Lectures
03:41:19

In this video we define the Laplace transform and discuss the types of differential equations we will be solving with it.

The Laplace Transform

05:17

Laplace Transform Example: Unit Step Function

04:50

Laplace Transform Example: First Derivative

04:27

In this video we find the Laplace transform of the second derivative of a function.

Laplace Transform Example: Second Derivative

03:10

Existence of the Laplace Transform

08:08

In this video we find the Laplace transform of the exponential function.

Laplace Transform Example: Exponential Function

02:27

Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine

08:18

In this video we define the inverse Laplace transform.

The Inverse Laplace Transform

03:58

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

Solving Differential Equations with Laplace Transform

11:10

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

Solving Differential Equations with Laplace Transform

08:30

In this video we look at some more techniques to invert Laplace transforms. We use partial fractions along with a table of Laplace transforms.

Partial Fractions to Invert Transforms

12:35

Here we learn a very important theorem, the First Translation Theorem. It is useful in finding inverse Laplace transforms that are translated/shifted.

First Translation Theorem

05:46

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

First Translation Theorem: Inverting Transforms

08:35

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

Completing the square is an important concept here.

First Translation Theorem: Inverting Transforms: Completing Square

07:59

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.

Second Translation Theorem

03:23

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

Piecewise Continuous Functions with Unit Step Functiond

06:59

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

Laplace Transform of Piecewise Continuous Functions

04:16

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

Laplace Transform of Piecewise Continuous Functions

05:29

In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.

Solving an IVP with a Piecewise continuous Non-homogenous Term

19:45

In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.

Solving an IVP with a Piecewise continuous Non-homogenous Term

13:12

Assignment 4

5 pages

In this video we look at a formula for the derivative of transforms and one application of it.

Derivatives of Transforms

07:42

In this video we find out how to take the Laplace Transform of periodic piecewise functions.

Laplace Transform of Piecewise Periodic Functions

05:30

In this video we solve an IVP with a periodic piecewise nonhomogenous term.

Solving an IVP with a Piecewise Periodic Non-homogenous Term

16:42

In this video we define the Dirac Delta Function which can be used to model impulses.

The Dirac Delta Function

10:55

In this video we solve an IVP with a delta function term.

Solving an IVP with a Delta Function Term

09:10

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

Solving an IVP with Multiple Delta Function Term

05:19

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.

The Convolution Theorem

06:58

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

Convolution Theorem: Finding Integral Solutions

04:18

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.

Convolution Theorem: Finding Integral Solutions

06:31

Assignment 5

7 pages

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Power Series Methods
20 Lectures
03:41:06

Print this document so you can follow along with the videos for this section.

Power Series Template Slides

12 pages

In this video we review quickly how to solve linear second order equations with constant coefficients. We then review some information about power series.

Review of Second Order Equations (Constant Coefficients) and Power Series

14:24

In this video we solve our first variable coefficient differential equation using power series.

Solving Airy's Differential Equation with Power Series Solution

22:02

In this video we learn to solve some differential equations in Maple and also find series solutions. We plot the Airy functions as well from the last video.

Plotting Solutions of Airy's DE and using Maple to find Series Solutions

19:49

In this video we solve another differential equation by finding a power series solution. We derive the recurrence relation for the coefficients. We also discuss more about initial conditions and how they determine the first two coefficients in the power series solution.We again use Maple to find the power series solutions as well. At the end we define ordinary points.

Finding a Power Series Solution, Using Maple as well, Ordinary Points

19:08

In this video we define ordinary points and singular points of second order linear differential equations. We complete one more example of finding power series solutions using the Chebyshev Differential equation. This video is continued in the next video.

Ordinary Points. Chebyshev's Differential Equation

20:13

Here we finish off the Chebyshev example from the previous video. We also quickly discuss Chebyshev polynomials.

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

We define singular and regular singular points. We begin to solve Euler's differential equation which leads to the indicial equation (we will be seeing a lot of it in this Section on series solutions.

Singular Points. Regular Singular Points. Euler's Differential Equation

15:12

This video continues from the previous one and we finish analyzing Euler's Differential Equation for the three different cases when considering roots of the indicial equation.

Euler's Differential Equation Continued

11:33

In this video we introduce the main theorem about finding Frobenius series solutions. It all revolves around the indicial equation and whether the roots differ by an integer or not. We also begin our first example of finding Frobenius series solutions when the roots differ by an integer.

Frobenius Series Solutions and Beginning of Example

13:15

We finish the example that was started in the last video. We find two guaranteed Frobenius series solutions in this example because the roots do not differ by an integer.

Frobenius Series Solution: Roots Differing by Non Integer

15:17

In this video we begin solving a differential equation where the roots of the indicial equation differ by an integer. In this case there is not always a Frobenius Series solution corresponding to the smaller root. In this example it turns out that there are two Frobenius solutions.

Frobenius Series: Roots Differing by an Integer - 2 Frobenius Solutions

04:58

This video takes off from where we left off in the last video. The roots to the indicial equation differ by an integer and it turns out there are two Frobenius series solutions. We see why in this video and how to solve such examples.

Roots Differing by an Integer - 1 Frobenius Solution Continued

18:39

In this example the roots of the indicial equation differ by an integer and there is only one Frobenius series solution corresponding to the larger root. We see why this is and learn how to solve such examples. We end the video with discussing how to find a second linearly independent solution. The answer is of course the reduction of order formula.

Frobenius Series: Roots Differing by an Integer- 1 Frobenius Solution

11:30

Quiz: Frobenius Series

1 page

When the roots to the indicial equation are equal there is only one Frobenius series solution. This also may happen when the roots differ by an integer. To find a second solution we can use reduction of order. In this video (and the next) we show how to use the method of reduction formula to find y2 when passing in a series solution for y1.

Method of Reduction with Frobenius Series

06:53

We finish the reduction question from the previous video and also show how to do one more from Example 8 in a previous video.

Method of Reduction with Frobenius Series Continued

22:22

Quiz: Frobenius and Reduction of Order

1 page

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Partial Differential Equations and Fourier Series
16 Lectures
04:19:42

Print this document so you can follow along with the videos for this section.

Partial Differential Equations and Fourier Series Template Slides

14 pages

Intro to PDEs

21:38

We begin our first example of solving the heat equation using separation of variables.

Separation of Variables: Heat Equation - Zero B.C.

18:58

We continue the separation of variables problem from the last video. An important concept that shows up here is the linear eigenvalue problem that leads to the eigenvalues and eigenfunctions.

Heat Equation - Zero B.C. - Sine Fourier Series

18:45

We continue on from the previous video. We discover the Sine Fourier series as we try to solve the heat equation.

Heat Equation - Zero B.C. - Sine Fourier Series Continued

22:52

We review the heat equation solution from the previous video and also demonstrate how to find the Sine Fourier series for a function. We plot the Fourier Series in Maple as well as the solution to the heat equation.

Sine Fourier Series Continued and Heat Equation

10:29

In this video we solve the heat equation zero flux boundary conditions. This leads to the cosine Fourier series which we also define. Separation of variables is used again to derive the eigenvalue problem.

Heat Equation - Zero Flux B.C. - Cosine Fourier Series

25:46

We solve the heat equation with periodic boundary conditions. This also leads to the full general Fourier series (involving both Cosine and Sine terms). This material is continued in the next video.

Heat Equation - Periodic B.C. General Fourier Series

18:23

A continuation of the previous video. We use orthogonality to solve for the coefficients in the Fourier series and also use Maple to animate the convergence of a Fourier Series.

Fourier Series Continued

17:01

Quiz - Fourier Series

1 page

We find the Fourier series for a function and also show how to use Maple to do some calculations. We also use a nice trick to show how Fourier series can be used to evaluate infinite series.

Fourier Series of Piecewise Continuous Function

23:27

Fourier Series - Convergence

11:42

We discuss even and odd functions as well as periodic extensions (even and odd). We show how the cosine and sine Fourier series are just special cases of the more general Fourier Series.

Cosine and Sine Fourier Series - Even and Odd Extensions

17:33

Separation of variables is used for linear homogenous PDEs and boundary conditions. Sometimes we can do a trick to remove inhomogeneous terms and then use separation of variables. We show this for the heat equation with a non homogenous term in the PDE.

Removing Inhomogeneous Terms in PDE: Heat Equation

22:36

We do the same trick as the last video to remove inhomogenous terms in the boundary conditions.

Removing Inhomogeneous Terns in Boundary Conditions: Heat Equation

07:45

In this video we solve the wave equation and look at some simulations in Maple.

Wave Equation

22:47

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Sturm-Liouville Eigenvalue Problems and Theory
9 Lectures
02:17:50

Slides for This Chapter

16 pages

We review some basics from linear algebra such as eigenvalues, eigenvectors, and symmetric matrices. We define self adjoint operators.

Self Adjoint Operators

23:28

We define the Sturm-Liouville (SL) operator and SL eigenvalue problems.

In this course we focus on Regular SL problems which we also define.

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

Learn how to solve nonhomogenous Sturm Liouville Problems.

Nonhomogenous Sturm Liouville Problem

20:39

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Nonlinear Systems
11 Lectures
03:04:36

Be sure to print off the slides so you can follow along with this section. They are added as resources to this video/lecture.

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

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Numerical Solutions to Differential Equations
21 Lectures
04:34:19

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

About the Instructor