I am a new college professor, teaching Differential Equations this fall for the first time, and I am finding myself needing to learn to use WolframAlpha and Mathematica by myself--as I was not taught DiffEq with the software. How can i get caught up to speed, at least with applications appropriate for Differential Equations, as quickly as possible?
First, I would get comfortable and understand the basics of Wolfram|Alpha. Most students will have used it and will try to use it in your course. Here are the example pages for Differential equations and integral transforms:
http://www.wolframalpha.com/examples/DifferentialEquations.html http://www.wolframalpha.com/examples/IntegralTransforms.html
Some of your students might subscribe to the pro version, which gives step-by-step instructions on how to solve problems. You probably want to subscribe yourself so you can see what they are seeing when they look at the step-by-step solutions.
Mathematica can be used in many ways to produce educational material for a class. Here's a video with a short introduction for teachers: https://www.wolfram.com/training/courses/edu001.html
One good example, done here in Mathematica, is the simple harmonic oscillator, using F = m a, where F = - k x
In[1]:= DSolve[m * x''[t] == -k * x[t], x[t], t] Out[1]= {{x[t] -> C[1] Cos[(Sqrt[k] t)/Sqrt[m]] + C[2] Sin[(Sqrt[k] t)/Sqrt[m]]}}