15
votes
How to prove Lambert's W function is not elementary?
In [Ritt 1948] p. 53 - 56, the method of J. Liouville is given for Kepler's equation. The same method can be applied to functions $f$ with $f(z)=A(z,e^z)$ ($A$ an algebraic function of two complex ...
- 1,053
7
votes
Accepted
Analytic solutions to algebraic differential equation
Your belief is not correct, as Robert Israel pointed in his comment. Same for
differential equations: $yy'-(3/2)z^2=0$ has a solution $y(z)=z^{3/2}$ which is not
analytic at $0$. The correct theorem ...
- 91.8k
5
votes
Accepted
Is there a theory of "elementary closed form solution" at the operator level for differential equations?
There are several theories which deal with this question.
One is the differential algebra, see, for example the little book
I. Kaplansky, Introduction to differential algebra,
Publ. de l'Institut de ...
- 91.8k
4
votes
Is every parametric function family, expressed as an elemetary function, a solution to an ODE with elementary functions?
Any elementary function solves an algebraic differential equation, i.e. where $F$ is a polynomial of its variables. I think you can find the result in: Eliakim Hastings Moore, "Concerning ...
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3
votes
Is every parametric function family, expressed as an elemetary function, a solution to an ODE with elementary functions?
To derive a differential equation, you differentiate
$y=G(x,c)$ with respect to $x$ and then eliminate $c$. This elimination process is not always possible with elementary functions. For example $y=e^...
- 91.8k
3
votes
Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?
For $m\neq -2$ the solution contains the Bessel function $J_{\nu}\left(\frac{2 \sqrt{a b}\, x^{1+m/2}}{m+2}\right)$ with index $\nu=\pm(m+2)^{-1}$, $\nu=-1\pm(m+2)^{-1}$, and $\nu=1\pm(m+2)^{-1}$. The ...
- 188k
2
votes
Is there work on differential Galois theory and infinite operators?
The March 2010 paper by Bernard Malgrange, Pseudogroupes de Lie et théorie de Galois différentielle, IHES/M/10/11 https://hal.archives-ouvertes.fr/hal-00469778 covers this case, I believe.
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2
votes
Non-linear first order ODE $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$
An analytic solution to the Abel equation of the second kind is claimed in this 2015 preprint by Rostami. As they say on Twitter, sharing does not constitute endorsement, but do check it out.
- 96.4k
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