Why Can't These Equations Be Solved Analytically?

[superg33k]

In my textbooks every now and again it says "these equations can't be solved analytically" or just "this can't be solved". For example my current book claims:

$$\frac{dx}{dt}=-kBe^{kz}\sin(kx-\omega t)$$, and
$$\frac{dz}{dt}=kBe^{kz}\cos(kx-\omega t)$$,

can't be solved analytically.

How do they know it can't be solved? I hope its the case that someone has proved it can't be solved, however I have never seen these proofs (I don't think). Is there an area of maths that that I can have a look at to understand more about how they make these statements? Or can anyone point me to some simple proof showing certain types of PDE's or polynomials or the above or something not too complicated that can't be solved?

Thanks for any help.

 

[micromass]

You are asking a deep question. It has indeed been rigorously proven that some polynomial equations/integrals/DE's can't be analytically solved. But the proof of this is by all means not easy.

To see why polynomials can't be solved in general, you must read a book on Galois theory. The book "a book on abstract algebra" is a very elementary introduction to Galois theory and provides a simple proof. But it still takes more than 200 pages before the proof can be given.
The book "Galois theory" by Stewart is a more thorough book.

To see why integrals can't be solved analytically, I must refer you to Liouville's theorem. See http://en.wikipedia.org/wiki/Liouville's_theorem_(differential_algebra)
The book "algorithms for computer algebra" by Geddes, Czapor, Labahn gives a nice proof of the fact without using too much abstraction.

In general, the solution to DE's and stuff requires differential Galois theory. See http://en.wikipedia.org/wiki/Differential_Galois_theory

 

[superg33k]

Thank you. I'll have a lot of fun exploring this. I'm sure I'll get lost quickly though.