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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!

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    $\begingroup$ It's not too surprising to me when something can't be solved. It's more surprising when things can be solved. $\endgroup$ – Jair Taylor Aug 7 '20 at 0:00
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    $\begingroup$ I am wondering if part of the problem is the size of the solution space. The powerset of the real numbers is absurdly large. $\endgroup$ – Q the Platypus Aug 7 '20 at 0:09
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    $\begingroup$ To add to the analogy with polynomials mentioned above: if we consider the simplest ODEs namely linear homogenous equations then the solutions of these are directly characterized by the roots of polynomial equations so in this case they go hand in hand. $\endgroup$ – Winther Aug 7 '20 at 1:03
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    $\begingroup$ What do you mean by "analytical solutions"? Or "finding" for that matter? Once you have shown that a solution exists and is unique, you have found the solution: namely, it is the only function satisfying your equation! Usually, if the equation is not too crazy, the solution will also be (mostly) analytical. I suspect what you meant to ask is "why are the solutions of differential equations usually not elementary functions". $\endgroup$ – tomasz Aug 7 '20 at 11:59
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    $\begingroup$ See math.stackexchange.com/questions/20578/… $\endgroup$ – lhf Aug 7 '20 at 14:47
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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.

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    $\begingroup$ One could go even further and say $f'(x) = f(x)$ doesn't have a solution either. Why, the exponential function is specifically defined to solve it... much like $\operatorname{erf}$ is specifically defined as an antiderivative of $x\mapsto e^{-x^2}$. In the same way, you could for lots of differential equations give the provably-existant&unique solution a name $\operatorname{foo}$ and then say “the equation can be solved exactly, the solution is $\operatorname{foo}$!” $\endgroup$ – leftaroundabout Aug 7 '20 at 15:12
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    $\begingroup$ For a nice article on the theorem of Liouville , see: Rosenlicht, M. (1972). Integration in finite terms. American Mathematical Monthly, Vol. 79, No. 9 (Nov., 1972), pp. 963-972. JSTOR, Google $\endgroup$ – Axion004 Aug 7 '20 at 16:01
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    $\begingroup$ @leftaroundabout I think you are using a very naive notion of "solution". If one uses "computable function", then the problems you name have a solution but OPs question is generally not solved. We can explicit write down a program that computes to arbitrary precision to result of $\exp(x)$ at arbitrary inputs $x$. The same is unknown for other kinds of integrals, but for $\exp$ and $erf$ it's basically all you'd want in practice, anyway. $\endgroup$ – WorldSEnder Aug 8 '20 at 13:21
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    $\begingroup$ @WorldSEnder If your notion of solution requires only that we can explicitly compute its value to arbitrary precision, then the existence and uniqueness result mentioned by leftaroundabout also gives a way to effectively compute it as well, namely using Picard's iteration. By the way, his notion of solution is not naive, it's just that you don't fully understand it. $\endgroup$ – BigbearZzz Aug 8 '20 at 15:46
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    $\begingroup$ @BigbearZzz I was thinking more about partial differential equations such as Navier-Stokes where uniqueness does not follow from Picard-Lindelöf. When Picard applies, "solved" in the sense I mentioned follows, as you correctly picked out. The notion of "solution" he alludes to is certainly not the correct one or even well-defined (hence naive), that's the whole point of his comment, as I understand it. I was merely pointing out a possible meaningful definition of "solution", I know there are many others. $\endgroup$ – WorldSEnder Aug 8 '20 at 16:37
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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.

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    $\begingroup$ In the context of your initial example, one should note that solving linear, constant coefficient, ordinary differential equations is already fully equivalent to finding roots of polynomials, and those are just a tiny and arguably one of the easiest subsets of all differential equations. $\endgroup$ – mlk Aug 7 '20 at 10:43
  • $\begingroup$ @mlk very true! I'll edit my answer to add this, thanks! $\endgroup$ – Riemann'sPointyNose Aug 8 '20 at 1:23
  • $\begingroup$ The collection of tricks for ODEs can be expressed as specializations of Lie symmetry reduction -- that is, the "trick" is to find a symmetry of the space of solutions. Of course, the set of ODEs having symmetry is a thin subset of the set of all ODEs. $\endgroup$ – Eric Towers Aug 10 '20 at 5:22
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Computable Functions are Rare

When stating mathematical problems, we usually state them in terms of elementary functions, but most certainly computable functions, because those are the only ones we know how to write down in finite space!

Because our brains can only explicitly conceptualize the computable functions, we have an innate bias towards thinking about these functions, and giving them a centrality within the world of numbers. When you read about diagonalization, it is tempting to think: "Non-computable numbers are such a hassle to build! Surely they must be rare!" But the reality is that the computable functions are the infinitely endangered species! There are only $\aleph_0$ such functions, but at least $\mathfrak c$ non-computable functions.

There are many ways to go from a computable number/function to a non-computable one (diagonalization being a well-known example, and the Halting Problem being another), but I would be surprised if anyone can name a "natural" problem which starts with a non-computable function/number and whose solution is computable (and by "natural", I mean one that isn't specifically contrived to do this).

An equation defines the intersection of two functions. If you take two arbitrary functions, what are the odds that those functions will intersect on one of the infinitesimally probable countable functions? This is why mathematicians are surprised when a result has a nice closed form. Usually, only trivial problems have this property.

Names Won't Save You

Of course, there is this business of deciding what is an "elementary function" or an "analytical solution". The answer is: "It doesn't matter." Those questions are totally irrelevant. Pick any finite set of problems that you like. Let us assign names to the solutions of those problems, regardless of whether they are computable or not. Now, we have greatly expanded the realm of "elementary functions". Awesome!!! We even did something amazing...we added some non-computable functions, which should really ramp-up our problem-solving power, right? Well, unless you got extraordinarily lucky, I'd bet against it.

An arbitrary non-computable function is garbage. It's less than worthless. While it is a solution to infinitely many problems, and thus expands your ability to write "closed form solutions" by a factor of at least $\aleph_0$, I'd wager, it is also not the solution (or remotely relevant) to a larger infinity of problems. Those problems require different non-computable functions than the ones you named.

Ok, ok...I'll let you cheat. I'll let you open up the toolbox and add some more functions. I didn't say how many you could add before, only that they had to be finite in number. You could have added a googol functions, I don't care. This time, I'm going to be really generous. I'm going to let you add an infinite number of non-computable functions, up to $\aleph_0$ of them!

Surely now we can write down nice "algebraic" solutions for most problems, given that we have beefed up our toolbox by a factor of infinity! But sadly, no. Our infinity isn't nearly big enough. No matter how clever you were at picking non-computable functions, there will still be infinitely many problems whose solution requires a non-computable function that you didn't choose.

You know what? I'm feeling generous. I feel bad, because you want mathematics to be nice and beautiful, and so far, it just looks like a giant mess. We tried to impose order by naming lots of solutions that didn't have names before. And as long as we stay under the $\aleph_0$ threshold, we can assign finite names to our "Augmented Algebraic Functions Toolbox". I'm going to do you one last favor. I'm going to let you add as many non-computable functions as you want! That should fix this problem once and for all, right?

Well, no. Now we have just traded one problem for another. If we simply add all functions $f: \mathbb R \rightarrow \mathbb R$ to our toolbox, we indeed capture a truly mind-bending number of functions, including more non-computable functions than you can shake a stick at! But the problem now is that we cannot name them! I mean, we can name them. We can put them in one-to-one correspondence with the reals. But, unfortunately, that means we cannot write most of them down! The only ones we can write down are the ones with a finite representation...and there's only $\aleph_0$ of those... D'oh!

And if we're being practical, nobody is going to read a paper which uses "elementary functions" with names that are 100 characters long. Probably 100 "elementary functions" is pushing the patience of most mathematicians. Unfortunately, $100 \lll \aleph_0 \lll \mathfrak c$. And so it goes...

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    $\begingroup$ This is a nice answer. Welcome to math stack exchange! $\endgroup$ – Don Thousand Aug 9 '20 at 13:54
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    $\begingroup$ @DonThousand: This is not a nice answer, but rather a severely misleading one. Whether or not a differential equation has an 'analytical solution' (solvable in terms of a certain class of standard functions) or not has absolutely nothing to do with computability. The error function, for example, is not an elementary function but it certainly is computable. So is the solution to essentially any differential equation with a computable expression and a computable initial boundary condition. $\endgroup$ – user21820 Aug 14 '20 at 12:13
  • $\begingroup$ As per what I just said, the paragraph starting with "An arbitrary non-computable function is garbage." is false garbage! $\endgroup$ – user21820 Aug 14 '20 at 12:15
  • $\begingroup$ This seems excessively ranty. $\endgroup$ – Pedro Tamaroff Dec 25 '20 at 23:54
  • $\begingroup$ The problems we write down and think about are also limited to $\aleph_0$ in theory and a small power of $10$ in practice. $\endgroup$ – J.G. Dec 26 '20 at 0:09
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I think you have kind of hit it on the head when you say that every time we can solve a differential equation, it is an algebraic coincidence. There is simply no good reason why any random equation should have a solution, let alone a nice or basic one.

The thought might come about as a result of having been taught, at school or early undergraduate level, certain methods that are useful for solving equations explicitly. But these are methods that are only applicable when the equations are already hand picked to be amenable to those methods.

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I think an analogy with computer science may provide some insight.

There are extremely simple programs that produce solutions of extraordinary complexity. The famous Rule 30 in cellular automata is the prime example: With a handful of bytes, one can write a deterministic program whose output is "as complex as possible," that is, it passes all measures of randomness. (It is actually amazing, to think of it.)

Yes, of course there are computer programs that produce simple outputs, but if we count only programs that produce distinct outputs as themselves distinct, then this number drops. One understand this fact in terms of Kolmogorov complexity.

As for mathematics, I think part of the problem is the finitude of the basis functions we accept as "solutions"... polynomials, trigonometric functions, exponential functions, special functions, etc. There are only a few of these. If we were to define by fiat new "special functions" that were the solutions to some of the current (hard) differential equations, we might think there are fewer "unsolved" equations.

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    $\begingroup$ In fact, many of the "special functions" are defined by fiat as solutions of differential equations. We've been using the "whatever that solution is, give it a name and pretend that it is now in the basket of elementary functions" strategy for a long time. It always surprised me that the Airy function was not one of these 18th and 19th century functions. $\endgroup$ – Eric Towers Aug 9 '20 at 21:06
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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.

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