Section 1: Introduction to Ordinary Differential
Equations. Modeling physical situations that exhibit exponential
growth and exponential decay. Plotting slope fields using the isocline
method. Sketching slope fields of autonomous differential
equations. Approximating solutions using Euler's method.
Lecture notes. 9 pages, last updated 1/21/11. Brannan/Boyce: Sections 1.1--1.3, 2.3, 8.1, supplemental material.
Lecture 1.1: What is a differential equation?
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Lecture 1.2: Plotting solutions to differential equations
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Lecture 1.3: Approximating solutions to differential equations
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Section 2: First Order Differential Equations. Solving 1st
order ODEs using separation of variables, the integrating factor
method, and variation of parameters. Structure of solutions to 1st
order linear ODEs, and connections to parametrized lines. Models of
motion with air resistance. Mixing problems. The logistic equation as
a population model.
Lecture notes. 21 pages, last updated 2/17/11. Brannan/Boyce: Sections 2.1--2.6.
Lecture 2.1: Separation of variables
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Lecture 2.2: Initial value problems
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Lecture 2.3: Falling objects with air resistance
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Lecture 2.4: Solving 1st order inhomogeneous ODEs
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Lecture 2.5: Linear differential equations
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Lecture 2.6: Basic mixing problems
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Lecture 2.7: Advanced mixing problems
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Lecture 2.8: The logistic equation
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Section 3: Second Order Differential Equations. Models that use
2nd order ODEs. Solving homogeneous linear 2nd order ODEs. Solving
inhomogeneous ODEs using the method of undertermined
coefficients. Simple harmonic motion. Harmonic motion with damping and
with forcing terms. The variation of parameters method for 2nd order
ODEs. Solving 2nd order non-constant coefficient ODEs. Cauchy-Euler
equations. The power series method, and the theorem of
Frobenius.
Lecture notes. 29 pages, last updated 2/17/11. Brannan/Boyce: Sections 4.1--4.7, 9.1--9.6.
Lecture 3.1: Second order linear ODEs
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Lecture 3.2: Equations with constant coefficients
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Lecture 3.3: The method of undetermined coefficients
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Lecture 3.4: Simple harmonic motion
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Lecture 3.5: Damped and driven harmonic motion
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Lecture 3.6: Variation of parameters
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Lecture 3.7: Cauchy-Euler equations
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Lecture 3.8: Power series solutions
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Lecture 3.9: The method of Frobenius
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Section 4: Systems of Differential Equations. Intro to
linear algebra: Adding and multiplying matrices. Writing systems of
linear equations with matrices, inverses and determinants of 2x2
matrices, eigenvalues and eigenvectors of 2x2 matrices. Using linear
algebra to solve systems of two 1st order linear ODEs x'=Ax; 3 cases
(i) real distinct eigenvalues, (ii) repeated eigenvalues, (iii)
complex eigenvalues. The SIR model in epidemiology.
Lecture notes. 26 pages, last updated 10/20/10. Brannan/Boyce: Sections 3.1--3.6, A.1.
Lecture 4.1: Basic matrix algebra
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Lecture 4.2: Eigenvalues and eigenvectors
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Lecture 4.3: Mixing with two tanks
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Lecture 4.4: Solving a 2x2 system of ODEs
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Lecture 4.5: Phase portraits with real eigenvalues
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Lecture 4.6: Phase portraits with complex eigenvalues
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Lecture 4.7: Phase portraits with repeated eigenvalues
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Lecture 4.8: Stability of phase portraits
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Lecture 4.9: Variation of parameters for systems
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Section 5: Laplace Transforms. Definition and properties of the
Laplace transform. Using Laplace transforms to solve ODEs. Using the
Heavyside function to express, and take the Laplace transform of,
piecewise continuous functions. Solving ODEs with discontinuous
forcing terms. Taking the Laplace transform of periodic
functions. Impulse functions and delta functions. Convolution.
Lecture notes. 21 pages, last updated 6/24/13. Brannan/Boyce: Sections 5.1--5.8.
Lecture 5.1: What is a Laplace transform?
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Lecture 5.2: Properties & applications of Laplace transforms
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Lecture 5.3: Discontinuous forcing terms
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Lecture 5.4: Periodic forcing terms
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Lecture 5.5: Impulse functions
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Lecture 5.6: Convolution
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Section 6: Fourier Series & Boundary Value Problems. Introduction to Fourier series -- derivation and computation. Even and odd functions, and Fourier cosine and sine series. Complex version of Fourier series. Parseval's
identity. Applications to summing series and to solving ODEs. Boundary values problems.
Lecture notes. 13 pages, last updated 12/9/11. Brannan/Boyce: Sections 10.1--10.3.
Lecture 6.1: Introduction to Fourier series
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Lecture 6.2: Computing Fourier series
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Lecture 6.3: Fourier sine and cosine series
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Lecture 6.4: Complex Fourier series
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Lecture 6.5: Applications of Fourier series
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Lecture 6.6: Boundary value problems
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Section 7: Partial Differential Equations The (1-dimensional)
heat, transport, and wave equations. Analysis of different boundary
conditions. Introduction to PDEs in higher dimensions. Harmonic
functions, Laplace's equation, and steady-state solutions to the heat
equation. Solving Laplace's equation, the heat equation, and the wave
equation in two dimensions.
Lecture notes. 23 pages, last updated 7/29/10. Brannan/Boyce: Sections 11.1--11.4, 11.6, 11.A, 11.B
Lecture 7.1: The heat equation
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Lecture 7.2: Different boundary conditions
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Lecture 7.3: The transport equation
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Lecture 7.4: The wave equation
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Lecture 7.5: Harmonic functions
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Lecture 7.6: Laplace's equation
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Lecture 7.7: The 2D heat equation
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Lecture 7.8: The 2D wave equation
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