Aren't most DEs not solvable analytically? i.e. only a tiny subset are neatly solvable - like precisely manufactured puzzles. In general, for any real problem y

Aren't most DEs not solvable analytically? i.e. only a tiny subset are neatly solvable - like precisely manufactured puzzles. In general, for any real problem you encounter "in the wild", numerical methods are the only way to find solutions (which are approximate). DE theory is still useful for designing frameworks within which numerical methods can operate. At least, that's my understanding.