# Questions tagged [differential-galois-theory]

The differential-galois-theory tag has no usage guidance.

The differential-galois-theory tag has no usage guidance.

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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'...

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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)...

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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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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 \\ $....

3
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1
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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
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2
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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. ...

2
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1
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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 ...

8
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1
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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
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2
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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 ...

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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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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-...

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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 $M$....