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Chapter 3
Systems of ODEs

 3.1 Introduction to systems of ODEs
  3.1.1 Exercises
 3.2 Matrices and linear systems
  3.2.1 Matrices and vectors
  3.2.2 Matrix multiplication
  3.2.3 The determinant
  3.2.4 Solving linear systems
  3.2.5 Computing the inverse
  3.2.6 Exercises
 3.3 Linear systems of ODEs
  3.3.1 Exercises
 3.4 Eigenvalue method
  3.4.1 Eigenvalues and eigenvectors of a matrix
  3.4.2 The eigenvalue method with distinct real eigenvalues
  3.4.3 Complex eigenvalues
  3.4.4 Exercises
 3.5 Two dimensional systems and their vector fields
  3.5.1 Exercises
 3.6 Second order systems and applications
  3.6.1 Undamped mass-spring systems
  3.6.2 Examples
  3.6.3 Forced oscillations
  3.6.4 Exercises
 3.7 Multiple eigenvalues
  3.7.1 Geometric multiplicity
  3.7.2 Defective eigenvalues
  3.7.3 Exercises
 3.8 Matrix exponentials
  3.8.1 Definition
  3.8.2 Simple cases
  3.8.3 General matrices
  3.8.4 Fundamental matrix solutions
  3.8.5 Approximations
  3.8.6 Exercises
 3.9 Nonhomogeneous systems
  3.9.1 First order constant coefficient
  3.9.2 First order variable coefficient
  3.9.3 Second order constant coefficients
  3.9.4 Exercises