Your first paragraph shows how much you do not understand scientific research. But it is an understandable misconception: most of the science that non-scientists are exposed to is the part that is closed and simple. The problems given in Physics exams are solvable, with the same techniques given in the course/class, the simplifying approximations fit, et cetera.

Nothing is farther from actual scientific research. Scientists don't research the known and solved problems - there is no research to do there. They research problems that are not yet solved, they try various theories, then retry, then fail, then try another approach, then fail, but in the process develop a technique that works somewhere else, so they sometimes go there, and so forth. I think I can recommend books such as "The Double Helix," that describe such research. Scientists work the same world that other researchers do: a complex, chaotic world, which is hard to predict, if at all possible. The only thing scientists have going for them, that non-scientists don't, is their analytical, quantitative methodology, and the fact that they _have_ solved in a closed manner a few significant private cases, which are widely implemented by engineers. A ridiculous question to any Physicist is: "How general is this closed form?" The answer is always to the effect of "as far as you can throw it."