Mathematics

Differential Equations

A course in differential equations, from first-order methods and linear equations to Laplace transforms, linear systems, nonlinear dynamics, numerical methods, boundary problems and introductory PDEs.

  • Intermediate
  • 78 lessons
  • Free
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About this course

A differential equation relates an unknown function to its rates of change. Differential equations describe how quantities change in time and across space, for example populations, temperatures, vibrating structures, electric circuits and chemical mixtures, and they are used throughout physics, engineering, biology and economics. This course starts with the language of differential equations and the meaning of a solution, then covers first-order methods and models, linear equations of higher order, oscillations and circuits, Laplace transforms, series solutions, linear systems, nonlinear dynamics, numerical methods, boundary and eigenvalue problems, Fourier series and introductory partial differential equations. It ends with Picard iteration and introductions to perturbation methods, delay equations and differential-algebraic equations.

The videos are coming out one at a time, in course order. Every lesson has its own page with precise definitions, a worked example for each key term and a few exercises with answers, so you can start reading now.

The reference sheet collects the definitions, solution methods, Laplace and Fourier transforms and theorems unit by unit, with the conditions each one needs and links to the lessons that explain them.

Single-variable calculus, algebra and trigonometry are enough to start. Complex numbers, matrices and eigenvalues, partial derivatives and Fourier series are introduced in the lessons where they are first needed, so Multivariable and Vector Calculus and Linear Algebra I are not required first.

Several topics of this course also appear elsewhere on the site. The First-Order Differential Equations and Fourier Series units of Calculus and the Further Differential Equations unit of Multivariable and Vector Calculus teach first-order equations and models, Fourier series, second-order linear equations, oscillations, series solutions, Laplace transforms and linear systems as part of those courses, and this course takes each of them further. The Linear Systems and Normal Modes unit explains the eigenvalues, eigenvectors and multiplicities it uses, and the Eigenvalues and Diagonalization unit of Linear Algebra I develops them more fully.

Good to know first

  • Single-variable differentiation and integration
  • Algebra and trigonometry

Course outline

1 of 78 lessons are on YouTube so far. Each lesson is one video. Open a unit to see its lessons, or follow a lesson for its definitions and examples.

  1. Language, Solutions and Modeling

    1 of 6 lessons on YouTube
    Open the Language, Solutions and Modeling unit page

    Equations, Solutions and Calculus Tools

    1. Differential Equations and Their LanguageOn YouTube

      Video 1

      differential equation, unknown function, ordinary differential equation, partial differential equation, order of an equation, dependent and independent variables, linear and nonlinear equations and homogeneous and nonhomogeneous linear equations

      Also uses derivatives of elementary functions.

    2. Solutions, Initial Conditions, and Boundary ConditionsComing soon

      Video 2

      explicit and implicit solutions, general and particular solutions, solution verification, domain of a solution, initial-value problem and boundary-value problem

      Builds on Differential Equations and Their Language. Also uses implicit differentiation.

    3. Calculus and Complex Numbers for Differential EquationsComing soon

      Video 3

      product and chain rules, antiderivatives and constants of integration, fundamental theorem of calculus, exponential and natural logarithm, complex numbers, Euler's formula and complex exponential

      Builds on Solutions, Initial Conditions, and Boundary Conditions.

    Direction Fields, Existence and Uniqueness

    1. Direction Fields and Solution CurvesComing soon

      Video 4

      direction field, local slope, isocline, solution curve and solution family

      Builds on Solutions, Initial Conditions, and Boundary Conditions and Calculus and Complex Numbers for Differential Equations.

    2. Existence, Uniqueness, and Maximal IntervalsComing soon

      Video 5

      local existence, local Lipschitz condition, uniqueness of solutions, maximal interval of existence, nonuniqueness and finite-time blowup

      Builds on Direction Fields and Solution Curves.

    Modeling and Units

    1. Modeling, Units, and NondimensionalizationComing soon

      Video 6

      state and parameter, balance law, units, dimensional consistency, nondimensionalization and characteristic scales

      Builds on Calculus and Complex Numbers for Differential Equations.

  2. First-Order Solution Methods

    7 lessons coming soon
    Open the First-Order Solution Methods unit page

    Separable and Linear Equations

    1. Separable EquationsComing soon

      Video 7

      separable equation, separation of variables, equilibria lost by division, implicit solution and solution interval

      Builds on Solutions, Initial Conditions, and Boundary Conditions, Calculus and Complex Numbers for Differential Equations and Existence, Uniqueness, and Maximal Intervals.

    2. Linear First-Order Equations and Integrating FactorsComing soon

      Video 8

      linear first-order equation, normalized linear form, integrating factor, derivative of a product, continuous coefficients and definite-integral solution formula

      Builds on Separable Equations.

    Exact Equations and Integrating Factors

    1. Exact Differential EquationsComing soon

      Video 9

      exact differential, exact equation, potential function, partial derivatives, test for exactness, local exactness and simply connected domain

      Builds on Calculus and Complex Numbers for Differential Equations and Separable Equations.

    2. Integrating Factors for Nonexact EquationsComing soon

      Video 10

      nonexact equation, integrating factor depending on x, integrating factor depending on y, quotient tests and domain restrictions

      Builds on Exact Differential Equations.

    Substitution Methods

    1. Homogeneous First-Order Equations and SubstitutionComing soon

      Video 11

      homogeneous first-order equation, scale invariance, ratio substitution, reduction to a separable equation and excluded solution branches

      Builds on Separable Equations.

    2. Bernoulli EquationsComing soon

      Video 12

      Bernoulli equation, power substitution, reduction to a linear equation, exceptional exponents and zero solution

      Builds on Linear First-Order Equations and Integrating Factors.

    3. Riccati Equations with a Known Particular SolutionComing soon

      Video 13

      Riccati equation, known particular solution, reciprocal substitution, linear equation for the correction, omitted particular branch and poles of solutions

      Builds on Linear First-Order Equations and Integrating Factors.

  3. First-Order Models and Qualitative Behavior

    5 lessons coming soon
    Open the First-Order Models and Qualitative Behavior unit page

    Phase Lines and Stability

    1. Autonomous Equations and Phase-Line StabilityComing soon

      Video 14

      autonomous equation, equilibrium solution, phase line, stable and unstable equilibria, asymptotic stability, semistability and derivative test for equilibria

      Builds on Direction Fields and Solution Curves, Existence, Uniqueness, and Maximal Intervals and Separable Equations.

    Growth, Decay and Population Models

    1. Exponential Growth, Decay, and Newton CoolingComing soon

      Video 15

      proportional rate of change, exponential growth and decay, half-life, time constant, Newton's law of cooling and constant ambient temperature

      Builds on Modeling, Units, and Nondimensionalization, Separable Equations and Linear First-Order Equations and Integrating Factors.

    2. Logistic Growth and HarvestingComing soon

      Video 16

      logistic model, carrying capacity, maximum natural growth rate, constant harvesting, threshold equilibrium, critical harvesting rate and physical domain

      Builds on Autonomous Equations and Phase-Line Stability and Exponential Growth, Decay, and Newton Cooling. Also uses partial fractions.

    Mixing and Motion Models

    1. Mixing and Compartment ModelsComing soon

      Video 17

      mass balance, amount and concentration, perfect mixing, variable volume and coupled compartments

      Builds on Modeling, Units, and Nondimensionalization and Linear First-Order Equations and Integrating Factors.

    2. Motion with Drag and Terminal VelocityComing soon

      Video 18

      sign convention for forces, linear drag, quadratic drag, terminal velocity and transient velocity

      Builds on Modeling, Units, and Nondimensionalization, Separable Equations and Linear First-Order Equations and Integrating Factors. Also uses Newton's second law of motion.

  4. Second-Order and Higher-Order Equations

    8 lessons coming soon
    Open the Second-Order and Higher-Order Equations unit page

    Linear Theory and Constant Coefficients

    1. Linear Equations, Superposition, and WronskiansComing soon

      Video 19

      linear differential operator, superposition principle, fundamental set of solutions, Wronskian, Abel's identity and regular interval

      Builds on Solutions, Initial Conditions, and Boundary Conditions, Calculus and Complex Numbers for Differential Equations and Existence, Uniqueness, and Maximal Intervals. Also uses two by two determinants.

    2. Second-Order Constant-Coefficient EquationsComing soon

      Video 20

      constant-coefficient homogeneous equation, characteristic equation, distinct real roots, repeated root, complex conjugate roots and real-valued solutions

      Builds on Linear Equations, Superposition, and Wronskians.

    3. Higher-Order Constant-Coefficient EquationsComing soon

      Video 21

      higher-order characteristic polynomial, root multiplicity, polynomial-exponential solutions, repeated complex roots and initial data for an nth-order equation

      Builds on Second-Order Constant-Coefficient Equations.

    Particular Solutions of Forced Equations

    1. Undetermined CoefficientsComing soon

      Video 22

      method of undetermined coefficients, complementary and particular solutions, trial function, overlap multiplicity and annihilator method

      Builds on Second-Order Constant-Coefficient Equations and Higher-Order Constant-Coefficient Equations.

    2. Variation of ParametersComing soon

      Video 23

      variation of parameters, fundamental pair, Wronskian denominator, equations for the parameter derivatives and particular solution by definite integrals

      Builds on Linear First-Order Equations and Integrating Factors, Linear Equations, Superposition, and Wronskians and Second-Order Constant-Coefficient Equations.

    Reduction of Order and Cauchy-Euler Equations

    1. Reduction of Order for Linear EquationsComing soon

      Video 24

      reduction of order, known solution, second independent solution, reduction of order formula and interval where the known solution is nonzero

      Builds on Linear Equations, Superposition, and Wronskians and Variation of Parameters.

    2. Cauchy-Euler EquationsComing soon

      Video 25

      Cauchy-Euler equation, equidimensional equation, power trial solution, logarithmic change of variable, repeated and complex exponents and singular point at the origin

      Builds on Second-Order Constant-Coefficient Equations and Reduction of Order for Linear Equations.

    3. Order Reduction in Nonlinear EquationsComing soon

      Video 26

      nonlinear order reduction, missing dependent variable, missing independent variable, velocity substitution, first integral and local coordinate restriction

      Builds on Separable Equations and Linear Equations, Superposition, and Wronskians.

  5. Oscillations, Response and Circuits

    4 lessons coming soon
    Open the Oscillations, Response and Circuits unit page

    Damped and Forced Oscillations

    1. Free and Damped Mechanical OscillationsComing soon

      Video 27

      mass-spring-damper model, viscous damping, underdamping, critical damping, overdamping, damped angular frequency and mechanical energy

      Builds on Modeling, Units, and Nondimensionalization and Second-Order Constant-Coefficient Equations.

    2. Forced Oscillations, Beats, and ResonanceComing soon

      Video 28

      forced oscillation, natural angular frequency, beats, undamped resonance, secular growth and damped amplitude peak

      Builds on Undetermined Coefficients and Free and Damped Mechanical Oscillations. Also uses trigonometric sum-to-product and product-to-sum identities.

    Frequency Response and Circuits

    1. Sinusoidal Steady State and Frequency ResponseComing soon

      Video 29

      sinusoidal steady state, complex gain, frequency response, amplitude ratio, phase lag and transient and steady-state response

      Builds on Calculus and Complex Numbers for Differential Equations and Forced Oscillations, Beats, and Resonance.

    2. RLC Circuits and Electromechanical AnalogiesComing soon

      Video 30

      series RLC circuit, Kirchhoff's voltage law, charge and current, resistance, inductance, capacitance and electromechanical analogy

      Builds on Free and Damped Mechanical Oscillations and Sinusoidal Steady State and Frequency Response.

  6. Laplace Transforms and Causal Response

    6 lessons coming soon
    Open the Laplace Transforms and Causal Response unit page

    The Laplace Transform and Its Inverse

    1. The Laplace TransformComing soon

      Video 31

      one-sided Laplace transform, exponential order, piecewise continuity, region of convergence and linearity of the Laplace transform

      Builds on Calculus and Complex Numbers for Differential Equations and Linear First-Order Equations and Integrating Factors. Also uses improper integrals.

    2. Inverse Laplace Transforms and Partial FractionsComing soon

      Video 32

      inverse Laplace transform, partial fractions, repeated poles, irreducible quadratic factor and completing the square

      Builds on The Laplace Transform.

    Initial-Value Problems and Switched Inputs

    1. Laplace Transforms of Derivatives and Initial-Value ProblemsComing soon

      Video 33

      Laplace transform of a derivative, integration by parts, initial-value terms, algebraic equation for the Laplace transform and initial-value problems by Laplace transform

      Builds on Second-Order Constant-Coefficient Equations, The Laplace Transform and Inverse Laplace Transforms and Partial Fractions.

    2. Step Functions, Time Shifts, and Periodic InputsComing soon

      Video 34

      Heaviside step function, time shift, piecewise inputs written with steps, second shifting theorem and Laplace transform of a periodic function

      Builds on Laplace Transforms of Derivatives and Initial-Value Problems.

    Convolution, Impulses and Transfer Functions

    1. Convolution and Linear-System ResponseComing soon

      Video 35

      causal convolution, convolution theorem, zero-state response, impulse-response kernel and Volterra integral equation

      Builds on Laplace Transforms of Derivatives and Initial-Value Problems and Step Functions, Time Shifts, and Periodic Inputs.

    2. Impulses, Transfer Functions, and PolesComing soon

      Video 36

      Dirac delta distribution, unit impulse, velocity jump at an impulse, transfer function, zero initial conditions, poles of a transfer function and bounded-input bounded-output stability

      Builds on RLC Circuits and Electromechanical Analogies, Step Functions, Time Shifts, and Periodic Inputs and Convolution and Linear-System Response.

  7. Series Solutions and Special Functions

    5 lessons coming soon
    Open the Series Solutions and Special Functions unit page

    Power-Series and Frobenius Solutions

    1. Power-Series Solutions at Ordinary PointsComing soon

      Video 37

      ordinary point, power-series solution, coefficient recurrence, index shift, initial coefficients and even and odd coefficient chains

      Builds on Calculus and Complex Numbers for Differential Equations and Linear Equations, Superposition, and Wronskians. Also uses power series and term-by-term differentiation.

    2. Analyticity and Convergence of Series SolutionsComing soon

      Video 38

      analytic coefficients, radius of convergence, normalized equation, nearest complex singularity and guaranteed versus exact radius

      Builds on Power-Series Solutions at Ordinary Points. Also uses the ratio test.

    3. Regular Singular Points and FrobeniusComing soon

      Video 39

      regular singular point, Frobenius series, indicial equation, roots differing by an integer, repeated indicial root and logarithmic second solution

      Builds on Cauchy-Euler Equations, Power-Series Solutions at Ordinary Points and Analyticity and Convergence of Series Solutions.

    Bessel and Legendre Functions

    1. Bessel's Equation and Bessel FunctionsComing soon

      Video 40

      Bessel's equation, order of a Bessel function, Bessel functions of the first and second kinds, behavior at the origin, half-order Bessel functions and radial problems

      Builds on Regular Singular Points and Frobenius.

    2. Legendre's Equation and Orthogonal PolynomialsComing soon

      Video 41

      Legendre's equation, polynomial termination, Legendre polynomials, standard normalization, orthogonality of Legendre polynomials and Legendre polynomials as eigenfunctions

      Builds on Power-Series Solutions at Ordinary Points and Regular Singular Points and Frobenius.

  8. Linear Systems and Normal Modes

    7 lessons coming soon
    Open the Linear Systems and Normal Modes unit page

    Systems and Eigenvalue Methods

    1. Systems and Conversion from Higher-Order EquationsComing soon

      Video 42

      state vector, first-order system, conversion of a higher-order equation to a system, vector initial data, matrix form of a linear system and autonomous and time-dependent systems

      Builds on Linear Equations, Superposition, and Wronskians.

    2. Real Eigenvalues and Linear-System SolutionsComing soon

      Video 43

      matrix multiplication, eigenvalue and eigenvector, eigenbasis, modal solution and constant coefficient matrix

      Builds on Systems and Conversion from Higher-Order Equations.

    3. Complex Eigenvalues and Planar Phase PortraitsComing soon

      Video 44

      complex eigenvector, real and imaginary part solutions, phase portrait, node, saddle, spiral and center

      Builds on Calculus and Complex Numbers for Differential Equations and Real Eigenvalues and Linear-System Solutions.

    4. Repeated Eigenvalues and Generalized EigenvectorsComing soon

      Video 45

      algebraic and geometric multiplicity, diagonalizable matrix, defective matrix, generalized eigenvector, Jordan chain and nilpotent part

      Builds on Real Eigenvalues and Linear-System Solutions.

    Matrix Exponentials and Forced Systems

    1. Matrix Exponentials and Fundamental MatricesComing soon

      Video 46

      matrix exponential, fundamental matrix, principal fundamental matrix, Liouville's formula and noncommuting coefficient matrices

      Builds on Real Eigenvalues and Linear-System Solutions, Complex Eigenvalues and Planar Phase Portraits and Repeated Eigenvalues and Generalized Eigenvectors. Also uses hyperbolic sine and cosine.

    2. Nonhomogeneous and Time-Dependent Linear SystemsComing soon

      Video 47

      variation of constants for systems, inverse fundamental matrix, nonhomogeneous linear system, time-dependent coefficient matrix and non-elementary integrals

      Builds on Variation of Parameters and Matrix Exponentials and Fundamental Matrices.

    Coupled Oscillators and Normal Modes

    1. Coupled Oscillators and Normal ModesComing soon

      Video 48

      coupled oscillators, mass matrix, stiffness matrix, generalized eigenvalue problem, normal modes and modal coordinates

      Builds on Free and Damped Mechanical Oscillations, Real Eigenvalues and Linear-System Solutions and Matrix Exponentials and Fundamental Matrices.

  9. Nonlinear Dynamics and Stability

    7 lessons coming soon
    Open the Nonlinear Dynamics and Stability unit page

    Phase Planes and Stability

    1. Nonlinear Phase Planes and NullclinesComing soon

      Video 49

      phase plane, vector field, nullcline, equilibrium point, trajectory, conserved quantity and predator-prey model

      Builds on Autonomous Equations and Phase-Line Stability and Systems and Conversion from Higher-Order Equations.

    2. Linearization and Local StabilityComing soon

      Video 50

      Jacobian matrix, linearization, hyperbolic equilibrium, local asymptotic stability, nonhyperbolic equilibrium and local versus global behavior

      Builds on Real Eigenvalues and Linear-System Solutions, Complex Eigenvalues and Planar Phase Portraits and Nonlinear Phase Planes and Nullclines.

    3. Lyapunov Functions, Energy, and InvarianceComing soon

      Video 51

      Lyapunov function, positive definite function, orbital derivative, Lyapunov stability, negative semidefinite derivative, LaSalle's invariance principle and invariant set

      Builds on Free and Damped Mechanical Oscillations and Linearization and Local Stability.

    Bifurcations and Limit Cycles

    1. One-Dimensional BifurcationsComing soon

      Video 52

      bifurcation parameter, bifurcation diagram, saddle-node bifurcation, transcritical bifurcation, exchange of stability, pitchfork bifurcation, reflection symmetry and supercritical and subcritical pitchforks

      Builds on Autonomous Equations and Phase-Line Stability and Linearization and Local Stability.

    2. Limit Cycles and Planar Limit-Set TheoremsComing soon

      Video 53

      periodic orbit, limit cycle, trapping region, omega-limit set, Poincaré-Bendixson theorem and Bendixson-Dulac criterion

      Builds on Nonlinear Phase Planes and Nullclines, Linearization and Local Stability and Lyapunov Functions, Energy, and Invariance. Also uses polar coordinates and the divergence theorem in the plane.

    3. Hopf Bifurcation and Oscillation OnsetComing soon

      Video 54

      Hopf bifurcation, eigenvalue crossing, polar normal form, transversality condition, nondegeneracy condition, first Lyapunov coefficient and supercritical and subcritical Hopf bifurcations

      Builds on Linearization and Local Stability, One-Dimensional Bifurcations and Limit Cycles and Planar Limit-Set Theorems.

    Sensitive Dependence and Chaos

    1. Chaos and Sensitive DependenceComing soon

      Video 55

      chaos, sensitive dependence on initial conditions, Lorenz system, volume contraction, attractor, largest Lyapunov exponent and forced Duffing equation

      Builds on Nonlinear Phase Planes and Nullclines, Limit Cycles and Planar Limit-Set Theorems and Hopf Bifurcation and Oscillation Onset.

  10. Numerical Methods and Reliable Computation

    6 lessons coming soon
    Open the Numerical Methods and Reliable Computation unit page

    Euler and Runge-Kutta Methods

    1. Euler's Method and Numerical ErrorComing soon

      Video 56

      Euler's method, step size, local truncation error, global error, order of accuracy and step refinement

      Builds on Calculus and Complex Numbers for Differential Equations, Direction Fields and Solution Curves and Existence, Uniqueness, and Maximal Intervals. Also uses Taylor's theorem with remainder.

    2. Improved Euler and Second-Order Runge-KuttaComing soon

      Video 57

      Heun's method, midpoint method, stage slopes, second-order Runge-Kutta methods and second-order global error

      Builds on Euler's Method and Numerical Error.

    3. Fourth-Order Runge-Kutta and Adaptive StepsComing soon

      Video 58

      classical Runge-Kutta method, RK4 stages, weighted average of slopes, fourth-order global error, embedded error estimate, adaptive step size and absolute and relative tolerances

      Builds on Improved Euler and Second-Order Runge-Kutta.

    Stiffness and Implicit Methods

    1. Stiffness, Stability, and Implicit MethodsComing soon

      Video 59

      stiffness, test equation, region of absolute stability, amplification factor, implicit Euler method, A-stability and numerical and physical stability

      Builds on Real Eigenvalues and Linear-System Solutions, Matrix Exponentials and Fundamental Matrices, Euler's Method and Numerical Error and Fourth-Order Runge-Kutta and Adaptive Steps.

    Numerical Boundary Problems and Verification

    1. Numerical Boundary-Value ProblemsComing soon

      Video 60

      shooting method, boundary residual, finite-difference method, tridiagonal system, conditioning and nonunique boundary solutions

      Builds on Second-Order Constant-Coefficient Equations, Euler's Method and Numerical Error and Fourth-Order Runge-Kutta and Adaptive Steps.

    2. Reliable Differential-Equation ComputationComing soon

      Video 61

      equation residual, initial and boundary checks, event detection, events versus singularities, convergence under refinement, independent checks and unit consistency

      Builds on Existence, Uniqueness, and Maximal Intervals, Modeling, Units, and Nondimensionalization, Fourth-Order Runge-Kutta and Adaptive Steps, Stiffness, Stability, and Implicit Methods and Numerical Boundary-Value Problems.

  11. Boundary Problems and Eigenvalue Methods

    5 lessons coming soon
    Open the Boundary Problems and Eigenvalue Methods unit page

    Solvability and Eigenfunctions

    1. Boundary-Value Problems and SolvabilityComing soon

      Video 62

      solvability of boundary-value problems, linear boundary map, kernel of the homogeneous problem, resonant forcing, compatibility condition and unique solvability

      Builds on Linear Equations, Superposition, and Wronskians and Second-Order Constant-Coefficient Equations.

    2. Eigenvalues and EigenfunctionsComing soon

      Video 63

      eigenvalue problem, eigenfunction, Dirichlet boundary conditions, Neumann boundary conditions, Robin boundary conditions, periodic boundary conditions, zero mode and normalization of eigenfunctions

      Builds on Boundary-Value Problems and Solvability.

    Sturm-Liouville Theory and Green's Functions

    1. Regular Sturm-Liouville ProblemsComing soon

      Video 64

      regular Sturm-Liouville problem, self-adjoint form, positive leading coefficient and weight, separated boundary conditions, weighted orthogonality and simple real eigenvalues

      Builds on Linear Equations, Superposition, and Wronskians, Legendre's Equation and Orthogonal Polynomials and Eigenvalues and Eigenfunctions.

    2. Green's Functions and the Fredholm AlternativeComing soon

      Video 65

      Green's function, integral kernel, jump condition at the source, eigenvalue obstruction, Fredholm alternative and Fredholm compatibility condition

      Builds on Variation of Parameters, Convolution and Linear-System Response, Boundary-Value Problems and Solvability and Regular Sturm-Liouville Problems.

    Fourth-Order Problems and Beams

    1. Higher-Order Eigenvalue Problems and Beam ModelsComing soon

      Video 66

      Euler-Bernoulli beam, fourth-order eigenvalue problem, clamped end, pinned end, free end and beam mode shapes

      Builds on Higher-Order Constant-Coefficient Equations, Coupled Oscillators and Normal Modes and Eigenvalues and Eigenfunctions.

  12. Fourier Methods and Partial Differential Equations

    8 lessons coming soon
    Open the Fourier Methods and Partial Differential Equations unit page

    Fourier Series

    1. Fourier Series and Orthogonal ExpansionsComing soon

      Video 67

      Fourier series, Fourier coefficients, orthogonality of sines and cosines, orthogonal projection, best mean-square approximation and Parseval's identity

      Builds on Calculus and Complex Numbers for Differential Equations, Eigenvalues and Eigenfunctions and Regular Sturm-Liouville Problems.

    2. Fourier Convergence, Gibbs Phenomenon, and Half-Range SeriesComing soon

      Video 68

      piecewise smooth function, pointwise convergence, average of one-sided limits at a jump, Gibbs phenomenon, odd and even extensions, half-range sine and cosine series and termwise differentiation

      Builds on Fourier Series and Orthogonal Expansions.

    PDE Classification and Transport

    1. PDE Types, Conditions, and Well-PosednessComing soon

      Video 69

      principal part, elliptic equation, parabolic equation, hyperbolic equation, compatibility of data, well-posedness, continuous dependence on data and backward heat equation

      Builds on Modeling, Units, and Nondimensionalization, Systems and Conversion from Higher-Order Equations and Boundary-Value Problems and Solvability.

    2. First-Order Transport and CharacteristicsComing soon

      Video 70

      transport equation, method of characteristics, characteristic curves, inflow boundary, variable speed, source term, inviscid Burgers' equation and weak solution

      Builds on Direction Fields and Solution Curves and PDE Types, Conditions, and Well-Posedness.

    Heat, Wave and Laplace Equations

    1. The Heat Equation and Separation of VariablesComing soon

      Video 71

      heat equation, thermal diffusivity, separation of variables for PDEs, Dirichlet and Neumann modes, conserved mean temperature, boundary lifting and diffusive decay

      Builds on Eigenvalues and Eigenfunctions, Fourier Series and Orthogonal Expansions, Fourier Convergence, Gibbs Phenomenon, and Half-Range Series and PDE Types, Conditions, and Well-Posedness.

    2. The Wave Equation, Normal Modes, and d'Alembert's FormulaComing soon

      Video 72

      wave equation, initial displacement and velocity, standing modes, fixed and free ends, traveling waves, d'Alembert's formula, domain of dependence and finite propagation speed

      Builds on Coupled Oscillators and Normal Modes, Fourier Series and Orthogonal Expansions, Fourier Convergence, Gibbs Phenomenon, and Half-Range Series, PDE Types, Conditions, and Well-Posedness and First-Order Transport and Characteristics.

    3. Laplace and Poisson EquationsComing soon

      Video 73

      Laplace's equation, Poisson's equation, harmonic function, separation in a rectangle and a disk, maximum principle, uniqueness for the Dirichlet problem, Neumann compatibility condition and uniqueness up to a constant

      Builds on Eigenvalues and Eigenfunctions, Fourier Series and Orthogonal Expansions, PDE Types, Conditions, and Well-Posedness and The Heat Equation and Separation of Variables.

    Fourier Transforms on the Whole Line

    1. Fourier Transforms and PDEs on Unbounded DomainsComing soon

      Video 74

      Fourier transform convention, Fourier transform of a derivative, inverse Fourier transform, Fourier transform of a Gaussian, heat kernel, unbounded domain and decay conditions

      Builds on The Laplace Transform, Fourier Series and Orthogonal Expansions, PDE Types, Conditions, and Well-Posedness and The Heat Equation and Separation of Variables. Also uses the Gaussian integral.

  13. Existence Constructions and Advanced Extensions

    4 lessons coming soon
    Open the Existence Constructions and Advanced Extensions unit page

    Existence by Successive Approximation

    1. Integral Equations and Picard IterationComing soon

      Video 75

      integral form of an initial-value problem, Picard iteration, successive approximations, sup norm, contraction estimate, fixed point and local versus global existence

      Builds on Existence, Uniqueness, and Maximal Intervals and Linear First-Order Equations and Integrating Factors.

    Perturbation, Delay and Differential-Algebraic Equations

    1. Perturbation Methods and Multiple ScalesComing soon

      Video 76

      small parameter, regular perturbation expansion, secular term, method of multiple scales, singular perturbation, boundary layer, composite approximation and interval of validity

      Builds on Second-Order Constant-Coefficient Equations, Free and Damped Mechanical Oscillations and Fourth-Order Runge-Kutta and Adaptive Steps.

    2. Delay Differential EquationsComing soon

      Video 77

      delay differential equation, history function, method of steps, infinite-dimensional state, characteristic roots of a delay equation and delay-induced instability

      Builds on Existence, Uniqueness, and Maximal Intervals, Linear First-Order Equations and Integrating Factors, Laplace Transforms of Derivatives and Initial-Value Problems and Integral Equations and Picard Iteration.

    3. Differential-Algebraic EquationsComing soon

      Video 78

      differential-algebraic equation, differential and algebraic variables, algebraic constraint, consistent initial values, hidden constraint, differentiation index and constrained pendulum

      Builds on Modeling, Units, and Nondimensionalization, Systems and Conversion from Higher-Order Equations and Nonhomogeneous and Time-Dependent Linear Systems.