As solutions of Differential Equations, what exactly are the functions sin(), cos(), exp() and ln() and well as how are rhey related? How do we know that they can be solutions to Differential Equations? Can somebody please prove or illustrate this to me? Preferably in terms of slope fields, but any way is alright.

I find it fascinating that we can just invent a function to be a solution to a DE. This is why I'm interested in such a construction.