Maybe solving some DEs analytically can help one build up intuition as to how the solution may behave, and some algebraic transformations may be useful, but maybe if the goal is for people to know if a solution makes sense then we should put more of an emphasis on the techniques to learn qualitatively about the solution (eg phase diagrams with isoclines and equilibrium points, perturbation analysis, various linearisations).

I wonder how important it is to focus on linearity. On the one hand, lots of equations aren’t linear but on the other hand, most equations are locally linear and knowing about linearity helps with Sturm-Liouville theory.