I would like to know if it's possible to solve differential equations and run them directly in either ArcMap or ERDAS Imagine, similar to the raster calculator in ArcMap.
There is nothing built into the software that can solve differential equations. I am assuming that your values are derived from spatial data or you would be posting this on another site. Your best bet (if you are tied to one of these software packages) is to write some code in ArcPy that pulls in your values and does the math. This Python library will help you immensely:
NumPy has been installed with ArcGIS since 9.2 and ESRI has some documentation on using it here: