Software

SageMath

• a Python-based front-end integrating many open-source packages (e.g. SciPy, Numpy, matplotlib, R, Mathjax, Tachyon, Maxima, ...)
• illustrations to Chapter 6, exercise 20, 21, and Chapter 9, exercise 22, 29.

Maxima

• designed primarily for *symbolic* computations
• similar functionality as Maple or Wolfram Mathematica
• typical problems: [DEMO] : derivatives and antiderivatives (primitive functions), limits, Taylor / Laurent expansions, partial fractions, residues, simplifications / expansions of expressions (polynomials, rational functions, trigonometric functions), matrix and vector calculations
• ODE problems: [DEMO] : linear equation with constant coefficients and given initial conditions; matrix exponential; Laplace transform and its inverse
  
• the plotdf command will plot the 2d phase picture with specific orbits.
• Maxima can also plot various 2d or 3d objects (see e.g. a tutorial [pdf]) or export the formulas in LaTex or console form (see the commands tex or grind).

Octave

• designed primarily for *numeric* computations
• similar to (and largely compatible with) Matlab
• typical problems: numerical integration, matrix calculations
• pictures and orbits can be computed / plotted
• simple graphs [DEMO]; numerical solutions to ODEs [DEMO]; numerical methods [DEMO];