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 ...
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^...
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 ...
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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