I'm looking for problems/lessons plans that could be used in a lower-division differential equations course that involve discerning properties of solutions of an equation, IVP, or BVP, without looking for an explicit/implicit solution (general or particular, given the context). Flow lines are an example of this, but I'm looking for something more advanced. One idea I've used is using first-order autonomous equations to figure out the dynamical properties of solutions for different initial conditions. I'm looking for similar ideas. Also: recommendations on how to present existence/uniqueness issues, besides showing a lot of examples, would be appreciated. (Boyce and DiPrima try to give a sketch of a proof of the basic existence/uniqueness result for first order IVPs. I wonder if this can be done without a course on analysis under your belt.)
Bonus: What about introducing group theoretic concepts at an early level? There are textbooks that claim to do this, but I wonder if this is as untenable as trying to teach measure theory in a calculus course.