Recrystallization of two unknown compounds?
Choosing a recrystallization solvent is as much an art as a science, even when you know what the substance is. Without knowing, it is a matter of making educated guesses. You want a solvent where it is much more soluble in the hot solvent than in the cold. Given that this is a lab exercise, we can assume that these are common substances. So, the one that dissolves in base is likely a carboxylic acid--perhaps benzoic acid or something close. That substance dissolves about 10 times as much in hot water as in cold - so I would try water first.  If it is too insoluble, then you could go to methanol, ethanol, etc., moving toward more nonpolar solvents until you get it to dissolve when heated. Then, hopefully it will crystallize out on cooling.  For the other substance, it is obviously soluble in dichloromethane and is not an acid. Perhaps it is non-polar. I would start with ethanol, and see how that works - if you need to go to a more non-polar solvent, toluene might be a good choice. Good luck!