Dynamical systems: Math 1b differential equations background.

I have been using the Strogatz textbook for teaching dynamical systems.  The course has multivariable calculus and linear algebra prerequisites.  Students might take the prerequisite courses different places.  For students who have taken Math 1b, AM/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b).

Student diff eq background from Math 1b:

  • Intro to diff eq
    • Write equations from a description of a model where the rate of change of a variable depends on the variable.
    • Determine whether a provided function is a solution to a differential equation.
    • Identify the order of a differential equation.
    • Find a solution to a linear, autonomous, first order differential equation.
  • Approximate solutions
    • Find equilibrium solutions to a differential equation.
    • Use slope fields to sketch approximate solutions.
    • Use Euler’s method to find approximate solutions.
    • Construct a differential equation that would have a particular slope field / a given solution behavior.
  • Separable differential equations
    • Identify whether a differential equation is separable.
    • Use separation of variables to solve separable differential equations.
  • Mass-spring systems
    • Find the characteristic equation for linear, constant coefficient, second order, homogeneous equations.
    • Learn Euler’s formula.
    • Use the characteristic equation to construct a general solution.
    • Identify whether a differential equation has periodic / oscillatory solutions.
    • Rewrite the second order equation as a system of two first order differential equations.
  • First order systems
    • Sketch a solution to a first order system in the phase plane.
    • Distinguish between competition and predator-prey relationships in interaction equations.
    • Draw nullclines in the phase plane.
    • Do a phase plane analysis: identify all nullclines, find all equilibria, orient each nullcline, draw arrows indicating the flow direction within each subregion of the phase plane.
    • Relate the values of dx/dt and dy/dt to the slope of the solution trajectory at a point.
    • Sketch several representative solution trajectories in a phase plane, constructing a phase portrait.
    • Work with the SIR model.