Why aren't all important differential equations solved and codified once and for all?

This is not a number question, and I hope it is on topic.

I always wondered how come, with modern supercomputers, the most important DE, especially the ones that refer to Physics, haven't been patiently solved one by one, and codified for reference, the way a century ago they did with sines, trig values and log tables.

Do you have an answer and an explanation, if the answer is negative?

The Lorenz Attractor is a set of differential equations. It's also chaotic, in the sense that slight perturbations of the initial conditions rapidly diverge to different results.

A pre-calculated "solution" for it is thus meaningless, because a table could only contain finitely many values but you cannot usefully interpolate between them.

This also applies to many real-world fluid dynamics and gravitational systems: no analytic solutions available, only numerical ones, and you have to be aware of the stability and convergence limitations of numerical solutions.

A large amount of effort has gone into algorithms for finding closed-form solutions to differential equations, and classification of differential equations of various types, and there has been a lot of progress. You don't need a supercomputer though: your average run-of-the-mill computer will do fine. Maple will find solutions of very many of the "important" differential equations that have closed-form solutions. It's a big field, though, and there's still lots of work to be done.

A good reason is that one couldn't carry tables weighting fifty megatons. (I don't mean paper tables, I mean fifty megatons of blue rays. But actually this figure is well below reality.)

A function such as the sine has a single independent variable, requiring say $10,000$ table entries. A function of two parameters would require $10,000^2$ of them. The solution of a PDE is a function of the independent variable, but also of the equation coefficients and the initial conditions. For instance, a tabulated second order homogeneous PDE could require $10,000^5$ $=100,000,000,000,000,000,000$ entries.

It is wrong to say that all important differential equations haven't been solved and codified. On the opposite, the solution of linear equations with constant coefficients is well established, both in theory and in practice (as it boils down to solving polynomial equations and linear systems).

For several other equation types, an integration procedure is known and can be fed to symbolic computation systems.

The modern way to "tabulate" solutions is by embedding symbolic and numerical computation algorithms in software such as Mathematica and others.