Is there a reason it is so rare we can solve differential equations?

Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some deeper reasoning behind why it is so rare to find solutions? Or is it just that every time we can solve differential equations, it is just an algebraic coincidence?

I reviewed the existence and uniqueness theorems for differential equations and did not find any insight. Nonetheless, perhaps the answer can be found among these?

A huge thanks to anyone willing to help!

Update: I believe I have come up with an answer to this odd problem. It is the bottom voted one just because I posted it about a month after I started thinking about this question and all you're inputs, but I have taken all the responses on this page into consideration. Thanks everyone!

Solution 1:

Let's consider the following, very simple, differential equation: $f'(x) = g(x)$, where $g(x)$ is some given function. The solution is, of course, $f(x) = \int g(x) dx$, so for this specific equation the question you're asking reduces to the question of "which simple functions have simple antiderivatives". Some famous examples (such as $g(x) = e^{-x^2}$) show that even simple-looking expressions can have antiderivatives that can't be expressed in such a simple-looking way.

There's a theorem of Liouville that puts the above into a precise setting: https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra). For more general differential equations you might be interested in differential Galois theory.

Solution 2:

Compare Differential Equations to Polynomial Equations. Polynomial Equations are, arguably, much, much more simple. The solution space is smaller, and the fundamental operations that build the equations (multiplication, addition and subtraction) are extremely simple and well understood. Yet (and we can even prove this!) there are Polynomial Equations for which we cannot find an analytical solution. In this way - I don't think it's any surprise that we cannot find nice analytical solutions to almost all Differential Equations. It would be a shock if we could!

Edit: in fact, users @Winther and @mlk noted that Polynomial Equations are actually "embedded" into a very small subsection of Differential Equations. Namely, Linear Homogeneous Constant Coefficient Ordinary Differential Equations, which take the form

$${c_ny^{(n)}(x) + c_{n-1}y^{(n-1)}(x) + ... + c_1y^{(1)}(x) + c_0y(x) = 0}$$

The solution to such an ODE in fact will utilise the roots of the polynomial:

$${c_nx^n + c_{n-1}x^{n-1} + ... + c_1x + c_0 = 0}$$

The point to make is that Differential Equations of this form are clearly just a teeny tiny small subsection of all possible Differential Equations - proving that both the solution space of Differential Equations is "much, much larger" than Polynomial Equations and already, even for such a small subsection - we begin to struggle (since any Polynomial Equation we cannot analytically solve will correspond to an ODE that we are forced to either (a) approximate the root and use it or (b) leave the root in symbolic form!)

Another thing to note is that solving equations in Mathematics is, in general, not a nice and easy mechanical process. The majority of equations we can solve usually do require methods to be built based on exploiting some beautiful, nifty trick. Going back to Polynomial Equations - the Quadratic Formula comes from completing the square! Completing the square is just a nifty trick, and by using it in a general case we built a formula. Similar things happen in Differential Equations - you can find a solution using a nice nifty trick, and then apply this trick to some general case. It's not as though these methods or formulas come from nowhere - it's not an easy process!

The last thing to mention in regards specifically to Differential Equations - as Mathematicians, we only deal with a very small subset of all possible Analytical Functions on a regular basis. ${\sin(x),\cos(x),e^x,x^2}$... all nice Analytical Functions that we have given symbols for. But this is only a small list! There will be an almighty infinite number of possible Analytical Functions out there - so it's again no surprise that the solution to a Differential Equation may not be able to be rewritten nicely in terms of our small, pathetic list.