This is a great webcast introducing you to Mathematica. Here is part 2. I suggest you look at it even if you know about Mathematica.
Marek Rychlik's JOde suite of programs for solving differential equations.
Here is a Mathematica Notebook for solving x dot = A x when A has distinct, real eigenvalues.
Here are the Mathematica notebooks I used to make the handout of Friday, Sept. 25: centerAndFocus.nb and JordanBockDynamics.nb. You can rotate the Jordan block phase portraits with the mouse.
Here is a Mathematica Notebook showing the flow of a linear ODE, done in class on Nov. 4.