# Questions tagged [differential-galois-theory]

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16
questions

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### Why the Riccati equation $\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}$ has an elementary solution "only" when $m=0$, $m=-2$, $m=4k/(2k\pm 1)$?

The special form of Riccati equation
$$
\frac{\mathrm{d} y}{\mathrm{d} x} =ax^{m}+by^{2}
$$ has been proved that it is solvable if and only if $m=0$, $m=-2$, $m=4k/(2k\pm 1)$.
The sufficiency is ...

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### Is there work on differential Galois theory and infinite operators?

I'm curious about differential Galois theory and I've noticed that everything I read covers only finite order operators (e.g. $L = Y^n + a_{n-1} Y^{n-1} + \dots + a_0 Y$). Has there been any work on ...

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### Is every parametric function family, expressed as an elemetary function, a solution to an ODE with elementary functions?

When given an ODE of the form $F(x, y, y', \ldots, y^{(n)}) = 0$, where $F$ is an elementary function, chances are that it has no solution of the form $y = G(x, c_1, \ldots, c_n)$, where $G$ is also ...

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### The asymptotic growth of codimension of range of polynomial differential equation on finite fields

Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question:
Let $P(x,y), Q(x,y)$ be two polynomials of ...

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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'...

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### 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)...

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### 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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### 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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### 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 ...

82
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### 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. ...

2
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### 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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### 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 ...

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### 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 ...

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### 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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### 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-...

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### 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 $M$....