I am a retired engineer studying (not in school) Maxwell's and Einstein's work at a beginner level. The concept of "solutions" to their equations comes up from time to time, and this puzzles me.

So I ask: What does it mean to have a "solution" to one of these equations? What is being solved for and why? I understand this question in the context of simple algebraic equations, but not for more complex systems such as Maxwell's wave equations and those of General Relativity. Can someone explain at appropriate level for me? Can someone point me to an example of a solution with obvious rational for it?

Thanks,

Lou.