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the menu is disabled
once the course is over
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Overview
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Syllabus
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Problem Sets
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Grades
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Overview and purpose of the course
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This course introduces a few mathematical techniques which are useful in describing the natural world. The emphasis will be to learn calculational tricks to solve differential equations that occur in the treatment of real, physical systems, rather than to learn rigorous proofs. More generally, the course is meant to give students a deeper sense of the relation of mathematics to physics.
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Lectures:
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11:30–12:20 Mondays, Wednesdays, and Fridays, Doherty Hall 1112
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Classes:
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9:30–10:20 Thursdays, Doherty Hall 1212
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Office hours:
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by arrangement
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Instructor:
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Hael Collins, Wean Hall Room 7414
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Grader:
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M. Sun, Doherty Hall MA 333
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Textbook:
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Edwards & Penney,
Differential Equations and Boundary Value Problems, Computing and Modeling,
Pearson, 2008.
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Schedule:
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Week I
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August 27–31
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General concepts about differential equations; first-order differential equations; direct integration
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Week II
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September 5–7
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Slope fields; existence and uniqueness; separable equations
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Week III
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September 10–14
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Implicit solutions; singular and general solutions; the exponential equation
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Week IV
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September 17–21
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First-order linear equations; substitution methods; Bernoulli equations; pursuit curves
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Week V
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September 24–28
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Exact differential equations; reducible second-order equations; differential equations as models; the logistic equation
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Week VI
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October 1–5
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Equilibria and stability; phase diagrams and bifurcation
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Week VII
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October 8–12
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Motion in a gravitational field with air resistance; variable mass (rocket problems)
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Week VIII
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October 15–18
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Approximation methods; Euler’s method
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Week IX
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October 22–26
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First Examination; the improved Euler method; the Runge-Kutta method; definitions and general properties of linear differential equations
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Week X
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Oct 29–Nov 2
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The Wronskian; solving homogeneous 2nd order linear equations with constant coefficients; complex numbers
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Week XI
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November 5–9
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The simple harmonic oscillator; damped oscillators; inhomogeneous equations; the method of undetermined coefficients
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Week XII
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November 12–16
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Driven, undamped oscillations; driven, damped oscillations; resonances; boundary value problems
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Week XIII
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November 19
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Second Examination
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Week XIV
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November 26–30
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Linear algebra: definition of a vector space; inner products; orthogonality; bases; linear transformations
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Week XV
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December 3–7
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Linear algebra: properties of matrices; transposing a matrix; Hermitian conjugation; matrix multiplication; eigenvalues and eigenvectors
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