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After studying both ODEs and PDEs, it did strike me that teaching this subject is incredibly counter-productive and outdated.The most important skill in this area is ability to compose differential equation from the description of the problem.That includes ability to correctly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