Blanchard, Devaney, and Hall 3rd edition (2006): Differential Equations. Sections 1.1-1.4, 1.8

Chapter 1: First order differential equations.  They present a goal: predicting a future value of a quantity modeled by a differential equation.

  • Section 1.1a.  Modeling via differential equations.  a: Introduce the idea of a model.  Distinguish between the independent variable (time), dependent variables (dependent on time) and parameters (don’t depend on time but can be adjusted).
  • Section 1.1b.  Modeling via differential equations.  b: Unlimited population growth.  P’ = k P is the equation (exponential growth).   Define first-order, ordinary differential equation, equilibrium solution, initial condition, qualitative analysis.  Introduce initial-value-problem, and solution.  Guess and check method of finding a solution.  Particular solution vs general solution.  Example comparing to United States population (annual census since 1790).
  • Section 1.1c.  Modeling via differential equations.  c: Logistic population growth.  Add a second assumption (at some level of population growth will become negative).  Logistic population model, nonlinear, equilibria.  They do a qualitative analysis and create approximate solutions.
  • Section 1.1d.  Modeling via differential equations.  d: Predator prey systems.  Add assumptions about fox and rabbit interactions.  They generate a first order system and define the solution to a system.
  • Section 1.1e.  Modeling via differential equations.  e: Analytic, qualitative, and numerical approaches.  Here they name that there are three approaches.
  • Section 1.2a.  Analytic technique: separation of variables.  a.  What is a differential equation and what is a solution?
  • Section 1.2b.  Analytic technique: separation of variables.  b.  Initial-value problems and the general solution.
  • Section 1.2c.  Analytic technique: separation of variables.  c.  Initial-value problems and the general solution.
  • Section 1.2d.  Analytic technique: separation of variables.  d.  Separable equations
  • Section 1.2e.  Analytic technique: separation of variables.  e.  Missing solutions
  • Section 1.2f.  Analytic technique: separation of variables.  f. Getting stuck
  • Section 1.2g.  Analytic technique: separation of variables.  g. A savings model
  • Section 1.2h.  Analytic technique: separation of variables.  h. A mixing problem
  • Section 1.3a.  Qualitative technique: slope fields.  a. The geometry of dy/dt = f(t,y)
  • Section 1.3b.  Qualitative technique: slope fields.  b. Slope fields
  • Section 1.3c.  Qualitative technique: slope fields.  c. Important special cases
  • Section 1.3d.  Qualitative technique: slope fields.  d. Analytic versus qualitative analysis
  • Section 1.3e.  Qualitative technique: slope fields.  e. The mixing problem revisited
  • Section 1.3f.  Qualitative technique: slope fields.  f. An RC circuit
  • Section 1.3g.  Qualitative technique: slope fields.  g. Combining qualitative with quantitative results
  • Section 1.4a.  Numerical technique: Euler’s method.  a. Stepping along the slope field
  • Section 1.4b.  Numerical technique: Euler’s method.  b. Euler’s method
  • Section 1.4c.  Numerical technique: Euler’s method.  c. Approximating an autonomous equation
  • Section 1.4d.  Numerical technique: Euler’s method.  d. A non-autonomous example
  • Section 1.4e.  Numerical technique: Euler’s method.  e. An RC circuit with periodic input
  • Section 1.4f.  Numerical technique: Euler’s method.  f. Errors in numerical methods
  • Leave existence and uniqueness (1.5), equilibria and the phase line (1.6), bifurcations (1.7), integrating factors (1.9) to a later course.
  • Section 1.8a.  Linear equations.  a.  Linear differential equations
  • Section 1.8b.  Linear equations.  b.  Linearity principles
  • Section 1.8c.  Linear equations.  c.  Solving linear equations
  • Section 1.8d.  Linear equations.  d.  Qualitative analysis
  • Section 1.8e.  Linear equations.  e.  Second guessing