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18.034

Honors Differential Equations

Undergraduate · Historical course

Spring 2009

Topics

  • Terminology and implicit solutions
  • Integration and solutions
  • Recitation 1
  • Fundamental principles
  • First-order linear equations
  • Recitation 2
  • Separable equations
  • Recitation 3
  • Linear fractional equations
  • Second-order linear equations
  • Mechanical oscillation
  • Recitation 4
  • Uniqueness and the wronskian
  • Separation and comparison theorems
  • Practice midterm 1
  • Midterm 1
  • Recitation 6
  • The maximum principle
  • Higher-order linear equations
  • Recitation 7
  • Solution bases
  • Recitation 8
  • Inhomogeneous equations
  • Stability
  • Recitation 9
  • Wellposedness; introduction
  • Recitation 10
  • Uniform convergence
  • Uniqueness and continuity
  • Practice midterm 2
  • Midterm 2
  • Remarks on wellposedness
  • Laplace transform
  • Transform and differential equations: generalized solutions, application to ODEs
  • Recitation 12
  • Step functions
  • Convolution
  • Recitation 13
  • The dirac distribution
  • Recitation 14
  • The transfer function and the pole diagram
  • Linear systems
  • Recitation 15
  • Eigenvalues and eigenvectors
  • Recitation 16
  • Complex solutions and the fundamental matrix
  • Practice midterm 3
  • Midterm 3
  • Repeated eigenvalues and the matrix exponential
  • Phase planes I
  • Phase planes II
  • Plane autonomous system
  • Stability and almost linear systems
  • Problems from ecology
  • Methods of Lyapunov
  • Nonlinear oscillations
  • The Poincare-Bendixson theorem
  • Recitation 18

Readings

  • Birkhoff, Garret, and Gian-Carlo Rota. Ordinary Differential Equations . 4th ed. New York, NY: Wiley, 1989. ISBN: 9780471860037.
    Listed reading
Spring 2004

Topics

  • Modeling and Terminology
  • Linear Differential Equations
  • Existence and Uniqueness of Solutions: Uniqueness
  • Existence and Uniqueness of Solutions: Picard Iterates
  • Extension of Solutions
  • Slope Fields
  • Qualitative Analysis
  • Approximate Numerical Solutions
  • Extra Topics and/or Review
  • Direction Fields
  • Inhomogeneous 2nd Order Linear ODE’s
  • Theory of 2nd Order Linear and Nonlinear ODE’s
  • Beats, Resonance, and Frequency Response Modeling
  • Extra Topics
  • Fourier Trigonometric Series
  • Half-range and Exponential Fourier Series
  • The Dirac Delta Function
  • The Laplace Transform: Solving IVP’s
  • Properties of the Transform
  • Convolution
  • Compartment Models
  • Eigenvalues, Eigenvectors and Eigenspaces
  • Homogeneous Linear Systems: Real Eigenvalues Case
  • Homogeneous Linear Systems: Complex Eigenvalues Case
  • Inhomogeneous Linear Systems: Exponentials of Matrices
  • Theory of General Linear Systems of ODE’s
  • The Fundamental Theorem
  • Autonomous Systems
  • Stability of Linear and Nonlinear Autonomous Systems
  • Conservative Systems
  • Limit Cycles and Planar Autonomous Systems

Readings

  • The readings referred to in the table are from the text: Borrelli, R. and C. Coleman. Differential Equations: A Modeling Perspective. Wiley Text Books, 2nd ed., New York, 2003. ISBN: 0471433322.
    Listed reading
  • SES #: 1 · TOPICS: Modeling and Terminology · READINGS: Chapter 1
    Reading assignment
  • SES #: 2 · TOPICS: Linear Differential Equations · READINGS: Chapter 2: Sections 2.1 and 2.2
    Reading assignment
  • SES #: 3 · TOPICS: Existence and Uniqueness of Solutions: Uniqueness · READINGS: Chapter 2: Sections 2.3 and A.1
    Reading assignment
  • SES #: 4 · TOPICS: Existence and Uniqueness of Solutions: Picard Iterates · READINGS: Chapter 2: Section A.2
    Reading assignment
  • SES #: 5 · TOPICS: Extension of Solutions · READINGS: Chapter 2: Section A.3
    Reading assignment
  • Separable Equations Chapter 2: Sections 2.4 and 2.5
    Listed reading
  • SES #: 7 · TOPICS: Qualitative Analysis · READINGS: Chapter 2: Section 2.8
    Reading assignment
  • SES #: 8 · TOPICS: Approximate Numerical Solutions · READINGS: Chapter 2: Page 122
    Reading assignment
  • SES #: 11 · TOPICS: Homogeneous 2nd Order Linear ODE’s with Constant Coefficients · READINGS: Chapter 3: Sections 3.1 and 3.2
    Reading assignment
  • Brief Review of Complex Numbers Chapter 3: Section 3.3
    Listed reading
  • SES #: 13 · TOPICS: Inhomogeneous 2nd Order Linear ODE’s · READINGS: Chapter 3: Sections 3.5 and 3.6
    Reading assignment
  • SES #: 14 · TOPICS: Theory of 2nd Order Linear and Nonlinear ODE’s · READINGS: Chapter 3: Sections 3.7 and 3.8
    Reading assignment
  • SES #: 15 · TOPICS: Beats, Resonance, and Frequency Response Modeling · READINGS: Chapter 4: Sections 4.1 and 4.3
    Reading assignment
  • SES #: 17 · TOPICS: Fourier Trigonometric Series · READINGS: Chapter 10: Sections 10.1 and 10.2
    Reading assignment
  • SES #: 18 · TOPICS: Half-range and Exponential Fourier Series · READINGS: Chapter 10: Section 10.3
    Reading assignment
  • SES #: 20 · TOPICS: The Dirac Delta Function · READINGS: Chapter 10: Page 328
    Reading assignment
  • SES #: 21 · TOPICS: The Laplace Transform: Solving IVP’s · READINGS: Chapter 5: Section 5.1
    Reading assignment
  • SES #: 22 · TOPICS: Properties of the Transform · READINGS: Chapter 5: Sections 5.2 and 5.3
    Reading assignment
  • SES #: 23 · TOPICS: Convolution · READINGS: Chapter 5: Section 5.4
    Reading assignment
  • Introduction to Linear Algebra Chapter 6: Section 6.1 and page 421
    Listed reading
  • SES #: 26 · TOPICS: Eigenvalues, Eigenvectors and Eigenspaces · READINGS: Chapter 6: Section 6.2
    Reading assignment
  • SES #: 27 · TOPICS: Homogeneous Linear Systems: Real Eigenvalues Case · READINGS: Chapter 6: Section 6.3
    Reading assignment
  • SES #: 28 · TOPICS: Homogeneous Linear Systems: Complex Eigenvalues Case · READINGS: Chapter 6: Section 6.4
    Reading assignment
  • SES #: 29 · TOPICS: Inhomogeneous Linear Systems: Exponentials of Matrices · READINGS: Chapter 6: Section 6.6
    Reading assignment
  • SES #: 30 · TOPICS: Theory of General Linear Systems of ODE’s · READINGS: Chapter 6: Section 6.8
    Reading assignment
  • SES #: 33 · TOPICS: The Fundamental Theorem · READINGS: Chapter 7: Section 7.1
    Reading assignment
  • Interacting Species Models Chapter 7: Sections 7.2 and 7.3
    Listed reading
  • SES #: 35 · TOPICS: Stability of Linear and Nonlinear Autonomous Systems · READINGS: Chapter 8: Sections 8.1 and 8.2
    Reading assignment
  • Lyapunov Functions Chapter 8: Sections 8.3 and page 508
    Listed reading
  • SES #: 37 · TOPICS: Limit Cycles and Planar Autonomous Systems · READINGS: Chapter 9: Sections 9.1 and 9.2
    Reading assignment
  • Borrelli, R., and C. Coleman. Differential Equations: A Modeling Perspective . Wiley Text Books, 2nd ed., New York, 2003. ISBN: 0471433322.
    Listed reading
Sources