But the irony is that in the broad scheme of things "interesting" exact solutions are very rare, almost freakish. The only systems we can reliably solve are linear :-). So while numerical approaches have weaknesses, they are the de-facto available tool in most cases.
What constitutes a closed-form expression depends on context/is up for debate:
- https://en.wikipedia.org/wiki/Closed-form_expression#Alterna...:
“Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.”
- https://mathworld.wolfram.com/Closed-FormSolution.html:
“For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.”
so that’s not a given.