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The wiki page on Numerical methods for ordinary differential equations states that "many differential equations cannot be solved using analysis." Is this literally true, or do they mean to say "it is not known how to solve many differential equations using analysis"? If it is literally true, how is that provable?

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    $\begingroup$ Note: "solved using analysis" in this context means "has a closed-form solution". $\endgroup$ – Omnomnomnom Sep 20 '16 at 16:00
  • $\begingroup$ What is the meaning of "having a closed form solution"? A polynomial with finite number of terms? A finite number of uses of elementary special functions ($\mathrm{exp}$, $\sin$, etc)? Or is it that the solution space smooth (problem well posed)? $\endgroup$ – Sean Lake Sep 20 '16 at 16:04
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In the same way that not all algebraic equations can be solved using algebra. For algebraic equations, you have Galois theory, and the solvability of an algebraic equation by algebraic means can be studies by studying the Galois group of that equation. Similarly, for differential equations one has differential Galois theory, and the deeply related Picard-Vessiot theory. Liouville's theorem also gives an answer that you might find useful.

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They literally mean that there does not exist a symbolic expression in terms of "elementary" functions.

For the simplest differential equation $$ y' = f(x) $$ it's clear that the solution for $y$ is simply the antiderivative of $f$. For some elementary functions $f$, it is possible to prove that there does not exist any elementary function whose derivative is $f$.

I have never studied the topic myself, but I know of several leads one could follow to study the subject:

That said, one can always extend the class of functions one considers. e.g. if you are really interested in a specific function $f$ and need to work with its antiderivative, then one gives the antiderivative a name and figures out its properties and how to work with and compute with it.

There is, I believe, a large amount of research (open and proprietary) in this direction; e.g. developing the classes of functions and algorithms used internally by symbolic math packages such as Mathematica.

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The question of whether a differential equation can be solved using "analysis" (or any particular class of functions) falls into the domain of differential Galois theory. For example, Liouville's theorem answers the question of "when is it possible to solve $y' = f(x)$ using analysis"?

For an example of an "unsolvable" differential equation, it is well known that $$ \frac{dy}{dx} = e^{-x^2} $$ has no closed-form solution.

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  • $\begingroup$ Why is $y=\mathrm{erf}(x)$ not closed-form, but $y=\sin(x)$ is? $\endgroup$ – Sean Lake Sep 20 '16 at 16:05
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    $\begingroup$ @SeanLake there is a common understanding of what usually qualifies as being "closed form". See the wiki page $\endgroup$ – Omnomnomnom Sep 20 '16 at 16:07
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The wikipedia page says that "analysis" in this context means "symbolic computation" - that is, a formula for the answer.

Most indefinite integrals can't be computed symbolically - see How can you prove that a function has no closed form integral? .

That means that often even the simplest differential equation - $dy/dx = f(x)$ - has no closed form solutions.

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