Course Information

Course

Subject:	Mathematics
Topic:	Differential Equations
Views:	17,542

Educator

Name:	Massachusetts Institute of Technology (MIT)
Type:	University

Visit Official Website

Support the MIT OpenCourseWare program

Donate to MIT

Instructor

Name

Gilbert Strang

Learn Differential Equations: Tutorials with Gilbert Strang and Cleve Moler

Video Lectures

Displaying all 68 video lectures.

Lecture 1 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	Response to Oscillating Input A linear equation with oscillating input has an oscillating output with the same frequency (and a phase shift).
Lecture 6 Play Video	Solution for Any Input To solve a linear first order equation, multiply each input by its growth factor and integrate those outputs.
Lecture 7 Play Video	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 Play Video	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 Play Video	Integrating Factor for Constant Rate The integrating factor multiplies the differential equation to allow integration.
Lecture 10 Play Video	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 Play Video	The Logistic Equation When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state.
Lecture 12 Play Video	The Stability and Instability of Steady States Steady state solutions can be stable or unstable â€“ a simple test decides.
Lecture 13 Play Video	Separable Equations Separable equations can be solved by two separate integrations, one in t and the other in y.
Lecture 14 Play Video	Second Order Equations For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency.
Lecture 15 Play Video	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 Play Video	Unforced Damped Motion With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation.
Lecture 17 Play Video	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 Play Video	Exponential Response — Possible Resonance Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.
Lecture 19 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	Laplace Transform: Second Order Equation The algebra problem involves the transfer function. The poles of that function are all-important.
Lecture 26 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	Two First Order Equations: Stability A second order equation gives two first order equations. The matrix becomes a companion matrix (triangular).
Lecture 31 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	Diagonalizing a Matrix A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Î› is the eigenvalue matrix.
Lecture 42 Play Video	Powers of Matrices and Markov Matrices Diagonalizing a matrix also diagonalizes all its powers.
Lecture 43 Play Video	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 Play Video	The Matrix Exponential The shortest form of the solution uses the matrix exponential multiplying the starting vector (the initial condition).
Lecture 45 Play Video	Similar Matrices If A and B are ""similar"" then B has the same eigenvalues as A.
Lecture 46 Play Video	Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.
Lecture 47 Play Video	Second Order Systems An oscillation equation has 2n solutions, n cosines and n sines. Those solutions use the eigenvectors and eigenvalues.
Lecture 48 Play Video	Positive Definite Matrices A positive definite matrix has positive eigenvalues, positive pivots, positive determinants, and positive energy.
Lecture 49 Play Video	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 Play Video	Boundary Conditions Replace Initial Conditions A second order equation can change from two initial conditions to boundary conditions at two points.
Lecture 51 Play Video	Laplace Equation Laplace's partial differential equation describes temperature distribution inside a circle or a square or any plane region.
Lecture 52 Play Video	Fourier Series A Fourier series separates a periodic function into a combination (infinite) of all cosine and since basis functions.
Lecture 53 Play Video	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 Play Video	Fourier Series Solution of Laplace's Equation Around every circle, the solution to Laplaceâ€™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 Play Video	Heat Equation The heat equation starts from a temperature distribution at t = 0 and follows it as it quickly becomes smooth.
Lecture 56 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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 Play Video	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.