Software

Maxima

http://maxima.sourceforge.io/
  • 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

http://octave.org
  • designed primarily for *numeric* computations
  • similar (and largely compatible) to 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];