TITLE: Is there a reason it is so rare we can solve differential equations?
QUESTION [65 upvotes]: 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!

REPLY [4 votes]: Original poster here! After gathering and the thoughts of everyone as well as my own, I believe I can answer this question in a way that satisfies you all.
Think of a polynomial function: $$p(t)=t^2-2t-24$$ $p$ may describe some scalar quantity, like population as time passes. When we speak about "solving" a polynomial, we tend to mean: "what is $t$ when $p=0$?" But notice, that's a completely separate desire that we had. Nothing is wrong with our equation, and it is not "unsolved". It's serving the purpose it said it would: you can input a time $t$ and it would tell you your population size at that time. Nothing about the setup guaranteed that we would be able to manipulate it somehow and find the INPUTS that give a desired OUTPUT (like zero). In this sense, the polynomial is a complete expression on it's own and is not "unsolved".
Now let's take a look at a differential equation: $$\frac{\partial{T}}{\partial{t}}=\alpha\frac{\partial^2{T}}{\partial{x^2}}$$ This is the equation for temperature for a one dimensional rod where $x$ is any point on a rod, $t$ is any point in time and $\alpha$ is a proportionality constant.
Here is what I want to emphasize: the equation is a complete statement. It says something true about how the temperature evolves along the rod. But because we as mathematicians and physicists want to know the function $T(x,t)$ so we can determine the full evolution of the rod from a starting state, we try to shift around the function in some way so that we can get a single $T$. But there was nothing about the differential equation that guaranteed that was possible. Sometimes it happens to work, sometimes it doesn't.
I think the rarity of our ability to find functions from DE's becomes a lot less odd when put this way: we have the desire to find the temperature function and are allowed to use all of the clues available. The differential equation is one clue we have to what it might be (in some situations it is our only clue).
It is only because we have the same desire so often, to find the function involved in our DE's, that we start saying that if we cannot find that information from the differential equation ALONE, it is not only unsolved because it is not in a form where we can simply read off the function, it is unsolvable.
Now for the few cases where the DE is the indeed the only clue we need to determine the function at it's heart, let's talk about how "simple" or "special" that function is.
Since this topic came up so frequently in the responses: as far as I can tell, there is no specific definition of a special function. For all our purposes in this post I think the following definition should be okay: a function is special if it is defined to be the answer to a problem. That problem could be a differential equation, like how the function $e^n$ is defined to be the solution to DE: $f(n)=\frac{df}{dn}$. We find ourselves needing to use the answer to that problem in so many places, it becomes convenient to make up a shorthand. Ultimately a special function is just a shorthand. My mistake was confusing rarer special functions for being more special or strange.