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2
votes
0answers
220 views

Can it be decided whether $\int\root 3 \of{\cos^2(t)}\,dt$ is expressible by elementary functions? [closed]

I would like to decide by methods of Differential Algebra whether the integral $\int\root 3 \of{\cos^2(t)}\,dt$ contrary to the output of CAS Mathematica Online Integrator might be expressible by ...
5
votes
2answers
144 views

Decidability of differential equations

Is there anything well-known about the algorithmic decidability of the satisfiability of an ODE $\dot{x}=f(x)$, $x: [0,1]\to R^n$ with an initial condition $x(0)=x_0$, given that $f(x)$ belongs to ...
0
votes
1answer
181 views

algebraic extensions of (differential) function fields

Let $K$ be a differential field with algebraically closed constant field $C$ (Think $\mathbb{C}(x)$ here). I am looking for an example of a simple algebraic extension $L = K[t]$, such, that $t' \notin ...
9
votes
2answers
967 views

Solvability in differential Galois theory

It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative. The proof I know goes as follows: Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
4
votes
0answers
268 views

Progress in J.F.Ritt's one problem?

In Ritt's classical book"Differential Algebra",he asked the following question(it can be found in page 177,problem 8): The decomposition problem: determine the number of times which the d.p.in a ...
5
votes
2answers
537 views

Differential ideal membership problem

We know that in the ordinary algebra ideal case, the ideal membership problem can be solved by the Grobner Base theory, then, is there a counterpart theory in the differential ideal case? To be ...
12
votes
2answers
782 views

Why do we need admissible isomorphisms for differential Galois theory?

Background: In Kaplansky's Introduction to Differential Algebra, an isomorphism between differential fields $K, L$ is defined to be admissible if $K,L$ are contained in a larger differential field ...