"We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission
of information. Information is an accidental feature of an elementary course in differential
equations"
And, if regard the material on differential equations as essentially nonsense, then good luck getting NSF grants for research in the subject!
Actually, can communicate a lot of good information in a course in differential equations, but to do this apparently need some exposure to some of the leading applications of differential equations.
Luckily, in this world, NSF grant writers are not typically undergraduate students, and there's plenty of money flowing towards ODE research (here are some examples: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1600381&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1408295&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1318480&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1418042&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1346876&Hist...) and of course there is much more cash going into PDEs.
I saw emphasis on control, optimization, numerical methods, and applications. Nice.
And, sure, PDE's stand to get more.
To a surprising extent, WHAT you learn about ODEs does not matter as much as developing enough familiarity with them that you can layer more complex stuff on top.
That said the course I took focused on systems of differential equations rather than second order differential equations. There is nothing like trying to do Laplace transforms of matrices of functions to demonstrate how important it is to avoid careless errors...
(On one memorable occasion I tried to solve the same problem 12 times and came up with 11 different answers - none of which were correct!)
That is where the wave equation comes in, in the first course on PDEs.