What are Differential Equations and how do they work?
After studying both ODEs and PDEs, it did strike me that teaching this subject is incredibly counter-productive and outdated.<p>The most important skill in this area is ability to <i>compose</i> differential equation from the description of the problem.<p>That includes ability to correctly identify and define initial conditions.<p>And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc).<p>Because in this way you can obtain  solution to any DE with any degree of precision (within reasonable assumptions about complexity of the problem).