> How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy

> How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy problems. Then you get into the real world data or a need for realistic modeling and it's numerical methods everywhere. And closed-form equations are themselves almost always simplified or abstracted models derived from real-world observations.