I had the experience of coming upon a differential equation, in the course of some research, which I could not solve explicitly. Mathematica choked on it and my boss and his office neighbor (both math PhD's) were unable to solve it explicitly. When I was about to give up set off to do it numerically my boss's neighbor suggested they call another fellow they both new. Two days later he delivered two neatly written sheets of paper with an explicit solution which featured a really novel (to all of us) substitution which facilitated the solution.

Now in the grand scheme of things the differential equation we were looking at might not be 'interesting' in the sense of being representative of a class of problems in a rich branch of math, but it was sure interesting to us, as it modelled the behavior of the system we were studying. We all were pretty sure there wasn't a closed form solution (but certainly weren't going to spend time proving that) and were pleasantly surprised. The solver did not get a co-author credit in the eventual paper, but he did get a shout out in a footnote.

Sure, but people manipulate them by hand plenty. The task is making them more amenable to solution by either A) converting it into one of the few closed form solutions, or B) converting it into a form more amenable to numerical solution.

But when there’s a closed-form solution it’s like God winking at you.