Questions tagged [differential-galois-theory]

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3 votes
1 answer
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Is there a theory of "elementary closed form solution" at the operator level for differential equations?

We begin by considering the usual general first order linear equation of the form $$ a_0 y' + a_1 y + a_2 = 0 $$ Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
1 vote
0 answers
85 views

About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation $$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$ it has a solution given by the Gauss's Hypergeometric function $$_2F_1(a,b;c;z)...
7 votes
0 answers
328 views

On the solvability of a nonlinear differential system

A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system ...
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5 votes
2 answers
579 views

Non-linear first order ODE $ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$

I am trying to solve an ODE which has the following form: $$ \dfrac{\mathrm{d}y}{\mathrm{d}x} = \frac{Axy \ + \ By^2 \ + \ Cy}{Dxy \ + Ey \ +\ Fx \ + G}$$ with an initial condition $y(x_0) = y_0 \\ $....
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3 votes
1 answer
361 views

Analytic solutions to algebraic differential equation

Dear Colleagues and Friends, Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive. I believe that it's known that if a ...
77 votes
2 answers
16k views

Why is differential Galois theory not widely used?

E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
  • 7,583
2 votes
1 answer
160 views

On a tower of strongly normal extensions

Where I could see the following statement? Let $K\subset L\subset M$ be a tower of the strongly normal extensions of differential fields. If $M$ is weakly normal over $K$, then $M$ is strongly ...
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8 votes
1 answer
283 views

Differential analogue of the newton polygon

While reading page 543-544 of Heun's differential equations, I came across what appears to be the differential analogue of the Newton polygon method. It reads as follows: Let us recall the ...
23 votes
2 answers
3k views

How to prove Lambert's W function is not elementary?

Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois theory extends that to Bessel functions, say. But what tools exist for ...
0 votes
1 answer
443 views

algebraic extensions of (differential) function fields

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

When is the monodromy group of a linear differential equation dense in the Galois group?

Given a system $Y'=A(t)Y$ with only regular singular points, then a theorem of Schlesinger says that the Zariski closure of the monodromy group is equal to the Galois group of the corresponding Picard-...
15 votes
2 answers
1k 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 $M$....
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