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18.03

Differential Equations

5-0-7 units. REST · Undergraduate

Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.

Spring 2010

Topics

  • Natural growth, separable equations
  • Direction fields, existence and uniqueness of solutions
  • Direction fields, integral curves, isoclines, separatrices, funnels
  • Numerical methods
  • Linear equations, models
  • Euler’s method; linear models
  • Solution of linear equations, integrating factors
  • First order linear ODEs; integrating factors
  • Complex numbers, roots of unity
  • Complex exponentials; sinusoidal functions
  • Linear system response to exponential and sinusoidal input; gain, phase lag
  • Complex numbers; complex exponentials
  • Autonomous equations; the phase line, stability
  • Linear vs. nonlinear
  • Solutions to second order ODEs
  • Modes and the characteristic polynomial
  • Good vibrations, damping conditions
  • Homogeneous 2nd order linear constant coefficient equations
  • Exponential response formula, spring drive
  • Exponential and sinusoidal input signals
  • Complex gain, dashpot drive
  • Operators, undetermined coefficients, resonance
  • Gain and phase lag; resonance; undetermined coefficients
  • Frequency response
  • LTI systems, superposition, RLC circuits.
  • Engineering applications
  • Fourier series: introduction
  • Fourier series
  • Operations on fourier series
  • Periodic solutions; resonance
  • Fourier series: harmonic response
  • Step functions and delta functions
  • Step response, impulse response
  • Step and delta functions, and step and delta responses
  • Convolution
  • Laplace transform: basic properties
  • Application to ODEs
  • Laplace transform
  • Second order equations; completing the squares
  • Laplace transform II
  • The pole diagram
  • The transfer function and frequency response
  • Linear systems and matrices
  • First order linear systems
  • Eigenvalues, eigenvectors
  • Eigenvalues and eigenvectors
  • Complex or repeated eigenvalues
  • Qualitative behavior of linear systems; phase plane
  • Linear phase portraits
  • Normal modes and the matrix exponential
  • Matrix exponentials
  • Nonlinear systems
  • Linearization near equilibria; the nonlinear pendulum
  • Autonomous systems
  • Limitations of the linear: limit cycles and chaos
  • Reviews

Readings

Prerequisites & course information

Prereq: None. Coreq: Calculus II (GIR) U (Fall, Spring) 5-0-7 units. REST Credit cannot also be received for 18.032 , CC.1803 , ES.1803

Sources