Learn Differential Equations: Tutorials with Gilbert Strang and Cleve Moler

Video Lectures

Displaying all 68 video lectures.
Lecture 1
Introduction to Differential Equations and the MATLAB® ODE Suite
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Introduction to Differential Equations and the MATLAB® ODE Suite
Gilbert Strang and Cleve Moler provide an overview to their in-depth video series about differential equations and the MATLAB® ODE suite.
Lecture 2
Overview of Differential Equations
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Overview of Differential Equations
Differential equations connect the slope of a graph to its height. Slope = height, slope = -height, slope = 2t times height: all linear. Slope = (height)^2 is nonlinear.
Lecture 3
The Calculus You Need
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The Calculus You Need
The sum rule, product rule, and chain rule produce new derivatives from known derivatives. The Fundamental Theorem of Calculus says that the integral inverts the derivative.
Lecture 4
Response to Exponential Input
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Response to Exponential Input
For a linear equation with exponential input from outside and exponential growth from inside, the solution is a combination of two exponentials.
Lecture 5
Response to Oscillating Input
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Response to Oscillating Input
A linear equation with oscillating input has an oscillating output with the same frequency (and a phase shift).
Lecture 6
Solution for Any Input
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Solution for Any Input
To solve a linear first order equation, multiply each input by its growth factor and integrate those outputs.
Lecture 7
Step Function and Delta Function
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Step Function and Delta Function
A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump.
Lecture 8
Response to Complex Exponential
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Response to Complex Exponential
For linear equations, the solution for a cosine input is the real part of the solution for a complex exponential input. That complex solution has magnitude G (the gain).
Lecture 9
Integrating Factor for Constant Rate
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Integrating Factor for Constant Rate
The integrating factor multiplies the differential equation to allow integration.
Lecture 10
Integrating Factor for a Varying Rate
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Integrating Factor for a Varying Rate
The integral of a varying interest rate provides the exponent in the growing solution (the bank balance).
Lecture 11
The Logistic Equation
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The Logistic Equation
When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state.
Lecture 12
The Stability and Instability of Steady States
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The Stability and Instability of Steady States
Steady state solutions can be stable or unstable – a simple test decides.
Lecture 13
Separable Equations
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Separable Equations
Separable equations can be solved by two separate integrations, one in t and the other in y.
Lecture 14
Second Order Equations
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Second Order Equations
For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency.
Lecture 15
Forced Harmonic Motion
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Forced Harmonic Motion
When the forcing is a sinusoidal input, like a cosine, one particular solution has the same form. But if the forcing frequency equals the natural frequency there is resonance.
Lecture 16
Unforced Damped Motion
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Unforced Damped Motion
With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation.
Lecture 17
Impulse Response and Step Response
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Impulse Response and Step Response
The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.
Lecture 18
Exponential Response — Possible Resonance
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Exponential Response — Possible Resonance
Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.
Lecture 19
Second Order Equations with Damping
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Second Order Equations with Damping
A damped forced equation has a sinusoidal solution with exponential decay. The damping ratio provides insight into the null solutions.
Lecture 20
Electrical Networks: Voltages and Currents
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Electrical Networks: Voltages and Currents
Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance).
Lecture 21
Method of Undetermined Coefficients
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Method of Undetermined Coefficients
With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.
Lecture 22
An Example of Undetermined Coefficients
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An Example of Undetermined Coefficients
This method is also successful for forces and solutions equal to polynomials times exponentials. Substitute into the equation!
Lecture 23
Variation of Parameters
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Variation of Parameters
Combine null solutions to find a particular solution for any right hand side. But it may involve a difficult integral.
Lecture 24
Laplace Transform: First Order Equation
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Laplace Transform: First Order Equation
Transform each term in the linear differential equation to create an algebra problem. You can transform the algebra solution back to the ODE solution.
Lecture 25
Laplace Transform: Second Order Equation
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Laplace Transform: Second Order Equation
The algebra problem involves the transfer function. The poles of that function are all-important.
Lecture 26
Laplace Transforms and Convolution
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Laplace Transforms and Convolution
When the input force is an impulse, the output is the impulse response. For all inputs the response is a ""convolution"" with the impulse response.
Lecture 27
Pictures of Solutions
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Pictures of Solutions
The direction field has an arrow with slope at each point coming from the differential equation. Arrows with the same slope lie along an ""isocline"".
Lecture 28
Phase Plane Pictures: Source, Sink, Saddle
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Phase Plane Pictures: Source, Sink, Saddle
Solutions to second order equations can approach infinity or zero. Saddle points have a positive and also a negative exponent or eigenvalue.
Lecture 29
Phase Plane Pictures: Spirals and Centers
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Phase Plane Pictures: Spirals and Centers
Imaginary exponents with pure oscillation provide a ""center"" in the phase plane. The point (position, velocity) travels forever around an ellipse.
Lecture 30
Two First Order Equations: Stability
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Two First Order Equations: Stability
A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular).
Lecture 31
Linearization at Critical Points
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Linearization at Critical Points
A critical point is a constant solution to the differential equation. The slope of the right hand side decides stability or instability.
Lecture 32
Linearization of Two Nonlinear Equations
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Linearization of Two Nonlinear Equations
With two equations, the two linearized equations use the 2 by 2 matrix of partial derivatives of the right hand sides.
Lecture 33
Eigenvalues and Stability: 2 by 2 Matrix, A
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Eigenvalues and Stability: 2 by 2 Matrix, A
Two equations with a constant matrix are stable (solutions approach zero) when the trace is negative and the determinant is positive.
Lecture 34
The Tumbling Box in 3-D
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The Tumbling Box in 3-D
A box in the air can rotate stably around its shortest and longest axes. Around the middle axis it tumbles wildly.
Lecture 35
The Column Space of a Matrix
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The Column Space of a Matrix
Capturing all combinations of the columns gives the column space of the matrix. It is a subspace (such as a plane).
Lecture 36
Independence, Basis, and Dimension
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Independence, Basis, and Dimension
Vectors are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension = number of basis vectors.
Lecture 37
The Big Picture of Linear Algebra
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The Big Picture of Linear Algebra
A matrix produces four subspaces: column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.
Lecture 38
Graphs
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Graphs
A graph has nodes connected by edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.
Lecture 39
Incidence Matrices of Graphs
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Incidence Matrices of Graphs
The incidence matrix has a row for every edge, containing -1 and +1 to show which two nodes are connected by that edge.
Lecture 40
Eigenvalues and Eigenvectors
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Eigenvalues and Eigenvectors
The eigenvectors remain in the same direction when multiplied by the matrix. Subtracting an eigenvalue from the diagonal leaves a singular matrix: determinant zero. An n by n matrix has n eigenvalues.
Lecture 41
Diagonalizing a Matrix
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Diagonalizing a Matrix
A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λ is the eigenvalue matrix.
Lecture 42
Powers of Matrices and Markov Matrices
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Powers of Matrices and Markov Matrices
Diagonalizing a matrix also diagonalizes all its powers.
Lecture 43
Solving Linear Systems
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Solving Linear Systems
An eigenvalue / eigenvector pair leads to a solution to a constant coefficient system of differential equations. Combinations of those solutions lead to all solutions.
Lecture 44
The Matrix Exponential
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The Matrix Exponential
The shortest form of the solution uses the matrix exponential multiplying the starting vector (the initial condition).
Lecture 45
Similar Matrices
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Similar Matrices
If A and B are ""similar"" then B has the same eigenvalues as A.
Lecture 46
Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors
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Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors
Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.
Lecture 47
Second Order Systems
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Second Order Systems
An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues.
Lecture 48
Positive Definite Matrices
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Positive Definite Matrices
A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy.
Lecture 49
Singular Value Decomposition (the SVD)
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Singular Value Decomposition (the SVD)
The SVD factors each matrix into an orthogonal matrix times a diagonal matrix (the singular value) times another orthogonal matrix: rotation times stretch times rotation.
Lecture 50
Boundary Conditions Replace Initial Conditions
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Boundary Conditions Replace Initial Conditions
A second order equation can change from two initial conditions to boundary conditions at two points.
Lecture 51
Laplace Equation
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Laplace Equation
Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region.
Lecture 52
Fourier Series
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Fourier Series
A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions.
Lecture 53
Examples of Fourier Series
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Examples of Fourier Series
Even functions use only cosines and odd functions use only sines. The coefficients in the Fourier series come from integrals.
Lecture 54
Fourier Series Solution of Laplace's Equation
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Fourier Series Solution of Laplace's Equation
Around every circle, the solution to Laplace’s equation is a Fourier series with coefficients proportional to r^n. On the boundary circle, the given boundary values determine those coefficients.
Lecture 55
Heat Equation
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Heat Equation
The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth.
Lecture 56
Wave Equation
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Wave Equation
The wave equation shows how waves move along the x axis, starting from a given wave shape and its velocity. There can be fixed endpoints as with a violin string.
Lecture 57
Euler, ODE1
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Euler, ODE1
Instructor: Cleve Moler

ODE1 implements Euler's method. It provides an introduction to numerical methods for ODEs and to the MATLAB suite of ODE solvers. Exponential growth and compound interest are used as examples.
Lecture 58
Midpoint Method, ODE2
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Midpoint Method, ODE2
Instructor: Cleve Moler

ODE2 implements a midpoint method with two function evaluations per step. This method is twice as accurate as Euler's method. A nonlinear equation defining the sine function provides an example. An exercise involves implementing a trapezoid method.
Lecture 59
Classical Runge-Kutta, ODE4
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Classical Runge-Kutta, ODE4
Instructor: Cleve Moler

ODE4 implements the classic Runge-Kutta method, the most widely used numerical method for ODEs over the past 100 years. Its major shortcoming is the lack of an error estimate. A simple model of the growth of a flame is an example that is used.
Lecture 60
Order, Naming Conventions
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Order, Naming Conventions
Instructor: Cleve Moler

The digits in the name of a MATLAB ODE solver reflect its order and resulting accuracy. A method is said to have order p if cutting the step size in half reduces the error in one step by a factor of two to the power p+1.
Lecture 61
Estimating Error, ODE23
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Estimating Error, ODE23
Instructor: Cleve Moler

ODE23 compares 2nd and 3rd order methods to automatically choose the step size and maintain accuracy. It is the simplest MATLAB solver that has automatic error estimate and continuous interpolant. ODE23 is suitable for coarse accuracy requirements.
Lecture 62
ODE45
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ODE45
Instructor: Cleve Moler

ODE45 is usually the function of choice among the ODE solvers. It compares 4th and 5th order methods to estimate error and determine step size.
Lecture 63
Stiffness, ODE23s, ODE15s
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Stiffness, ODE23s, ODE15s
Instructor: Cleve Moler

A problem is said to be stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results. The flame model demonstrates stiffness.
Lecture 64
Systems of Equations
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Systems of Equations
Instructor: Cleve Moler

An ODE involving higher order derivatives is rewritten as a vector system involving only first order derivatives. The classic Van der Pol nonlinear oscillator is provided as an example. The VdP equation becomes stiff as the parameter is increased.
Lecture 65
The MATLAB ODE Suite
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The MATLAB ODE Suite
Instructor: Cleve Moler

The MATLAB documentation provides two charts summarizing the features of each of the seven functions in the MATLAB ODE suite.
Lecture 66
Tumbling Box
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Tumbling Box
Instructor: Cleve Moler

Throw a rectangular box with sides of three different lengths into the air. You can get the box to tumble stably about its longest axis or its shortest axis. But if you try to make it tumble about it middle axis, you will find the motion is unstable.
Lecture 67
Predator-Prey Equations
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Predator-Prey Equations
Instructor: Cleve Moler

The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. The program ""predprey"" studies this model.
Lecture 68
Lorenz Attractor and Chaos
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Lorenz Attractor and Chaos
Instructor: Cleve Moler

The Lorenz chaotic attractor was discovered by Edward Lorenz in 1963 when he was investigating a simplified model of atmospheric convection. It is a nonlinear system of three differential equations. The program ""lorenzgui"" studies this model.