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Introduction to Differential Equations

  • Definitions and terminology
  • Initial-value problems
  • Differential equations as mathematical models

First-Order Differential Equations

  • Solution curves without a solution; direction fields, autonomous first-order differential equations
  • Separation of variables
  • Linear equations
  • Exact equations
  • Solutions by substitutions
  • Numerical methods; Euler's method, numerical solvers

Modeling with First-Order Differential Equations

  • Linear models; exponential growth and decay, Newton's law of cooling, mixture problems
  • Non-linear models; logistic growth, chemical reactions
  • Systems of differential equations; radioactive series, mixtures, predator-prey models

Higher-Order Differential Equations

  • Linear differential equations; initial-value and boundary-value problems, homogenous equations, non-homogeneous equations
  • Reduction of order
  • Homogenous linear equations with constant coefficients
  • Undetermined coefficients; supposition approach, annihilator approach
  • Variation of parameters
  • Cauchy-Euler equation
  • Solving systems on linear equations using elimination
  • Non-linear differential equations

Modeling with Higher-Order Differential Equations

  • Linear models with initial value problems; spring/mass systems with free undamped motion, free damped motion, and driven motion
  • Linear models with boundary value problems
  • Nonlinear models

Series Solutions of Linear Equations

  • Review of power series
  • Solutions about ordinary points
  • Solutions about singular points
  • Special functions; Bessel's Equation, Legendre's Equation

The Laplace Transform

  • Definition of the Laplace transform
  • Inverse transforms and transforms of derivatives
  • Operational properties of the transform
  • The Dirac Delta function

Systems of Linear First-Order Differential Equations

  • Preliminary theory; supposition principle, independence, general solutions
  • Homogenous linear systems; distinct real eigenvalues, repeated eigenvalues, complex eigenvalues
  • Nonhomogeneous linear systems; undetermined coefficients, variation of parameters
  • Matrix exponentials

Numerical Solutions of Ordinary Differential Equations

  • Euler's method
  • Improved Euler's method
  • Runga-Kutta methods (RK4)