Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certai

Oh it is special precisely because of that! We can almost always get the answer with numerical methods. Yet, numerical solutions are good almost only for certain values of parameters. Analytic solutions on the other hand provide more insight. It can be differentiated, integrated, compared and maybe (big maybe) transform to some other/more familiar form. Finally it can be connected to whole body of mathematical knowledge. Analytic solutions are very satisfying for that reasons. Closed solutions are small, very narrow subset of analytic ones, that’s why they feel special. But I suppose they become special only after some time spent with problems where closed form solutions are rare and unexpected. I suppose high school/college mathematics makes us way too used to rare special cases, such as closed form solutions.