>The last two texts I've taught second semester calculus from have >included a chapter on Ordinary Differential Equations.
> Is >this >becoming a common practice? Is it good? Does it undermine or bolster the >ODE course? Is it too shallow a treatment? It certainly seems an important >topic for many of those majors which are not required to take ODE's as >a separate course, but do require two semesters of calculus. > At Northeastern for years we included a unit on differential equations in our first year course for engineers. This was partially because the engineering departments did not want their students to wait until the ODE course to see some of this stuff. it also gave us a chance to have the students do some serious modeling using differential equations. Students did a project on topic they were interested in which could be attacked by first order ODEs.
I think there are a lot of advantages of mixing first order ODEs in the calculus curriculum. It does get the students thinking about the meaning of the derivative and the integral in a different way in the first year courses. If you talk about exact equations in the context of gradient fields, it adds a geometric dimension to the exact equations usually missing from most ODE treatments. Having talked about slope fields in the first year course, students in multivariable calculus realize quickly that vector fields are very similar to slope fields, so it's not too surprising that there is a connection between vector fields and differential equations.
Pushing the "cookbook" topics into earlier calculus courses means that in the differential equations course you can focus on topics which have more coherence like second order equations and systems.