Differential Equations Home

Packet 01

• Complex numbers
  • Arithmetic
  • Exponents
  • Polar format
  • Conjugate
  • Division
  • Roots

Packet 02

• What are differential equations?
  • Examples of differential equations
  • Concepts of differential equations
• First-order equations
  • Direct integration
  • Separable equations: substitution
  • Linear equations: product rule
  • Exact equations: gradients
• Problems

Packet 03

• Transformation and reduction
  • Homogeneous equations: conversion to separable
  • Integrating factors: conversion to exact
  • Reduction of order
  • Ad hoc transformations
• Families of integral curves
• Exercises B
• Problems

Packet 04

• Functions and differential equations
  • Excursus: functions, derivatives, integrals
  • Initial value problems
  • Functional viewpoint and IVPs: pathological phenomena
  • Picard-Lindelöf Theorem: local well-posedness
  • Picard iteration
• Problems

Packet 05

• Applications: qualitative behavior
  • Paths in the plane
  • Growth and decay
  • Autonomous equations
• Problems

Packet 06

• Numerical techniques
  • Euler’s method
  • Heun’s method
  • Runge-Kutta methods
  • Adams method: multistep: polynomial fitting
  • Errors
  • Adaptive methods
  • Stability and stiff problems
• Problems

Packet 07

• Linear equations in higher order
  • Second-order, constant coefficients
• Problems

Packet 09

• Inhomogeneous theory
• Variation of parameters
  • Discovering a second solution
  • Particular solution of inhomogeneous
• Applications
  • Damped harmonic oscillator
  • Driven harmonic oscillator
  • Driven, damped harmonic oscillator
  • Series RLC circuit
  • MIT Mathlet links
• Problems

Packet 10

• Power series solutions to ODEs
  • Review of power series
  • Power series solutions: basic technique
  • Second-order linear equations
• Problems
  • Easier Problems
  • Harder Problems

Packet 11

• Problems
  • Easier Problems
  • Harder Problems

Linear algebra tutorial

• Basic vector algebra
• Spans
• Matrices: linear actions on vectors
• Determinant
• Independence
• Matrix algebra: composition as multiplication
• Eigenvectors
• Finding eigenvalues and eigenvectors
• Matrix calculus

Packet 12

• Linear systems I: matrix descriptions
  • Systems from higher order equations
  • Homogeneous system
• Problems
  • Easier Problems
  • Harder Problems

Packet 13

• Linear systems: homogeneous, constant coefficients
  • Eigenvector approach
  • Complex eigenvalues
  • Insufficient eigenvectors
  • Matrix exponential approach
• Problems
  • Easier Problems
  • Harder Problems

Packet 14

• Fourier series
  • Fourier series basics
  • Fourier series: orthogonality relations
• Boundary value problems
  • Eigenfunctions
• PDE: Wave equation
  • Traveling waves
  • Standing waves
• Problems
  • Easier Problems
  • Harder Problems

Final exam study

Differential Equations - Study Guide - Final Exam