Questions tagged [differential-algebra]

Differential algebra is the study of differential rings and fields and related structures.

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311
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7answers
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How can you prove that a function has no closed form integral?

In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/...
0
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0answers
7 views

Finding the particular integral of sin function when denominator became zero on putting a and b in 2nd order differential equations

Originally, Ques is find the complete integral of (D-3D'-2)²z= 2e^(2x) sin(y+3x). After solving it I found the CF. But while finding the PI, I came to an equation i.e. PI = 2e^(2x) {1/(D-3D')²}sin(y+...
3
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1answer
136 views

Explain Application of Risch's Structure Theorem for Elementary Functions

Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples? The Structure Theorem for Elementary Functions: "Let $(\mathfrak{E},Y)$...
4
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0answers
72 views

How to apply Risch's algorithm to $\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x$?

It is known that $$\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x=$$ $$-\frac{1}{6}\log\Big((2x^4-10x^3+24x^2-28x+14)\sqrt{x^4-2x^3+3x^2+4x+1}-2x^6+12x^5-36x^4+56x^3-42x^2+13\Big)+C.$$ To get ...
4
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0answers
143 views

Prove that $\int \sin(x^2)dx$ is not elementary

See edit It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series. However, is ...
7
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1answer
127 views

Can we find a simple basis for the cokernel of this derivation?

Let $K$ be a field of characteristic zero. Let $R$ be the $K$-algebra $K[x_0,x_1,\ldots]$ of polynomials in countably infinitely many variables. Consider the $K$-linear derivation $\delta:R\to R$ ...
1
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1answer
106 views

Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page $45$, definition $5.2$, author introduce the notion of $\mathfrak{g}$-...
0
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1answer
35 views

Bounded function and Derivatives [closed]

Does eventually all the successive derivatives of a bounded functions become bounded if one of them becomes bounded?(for entire number line case)
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0answers
31 views

A priori structural identifiability analysis of a system of ordinary differential equation using differential algebra

Consider an ordinary differential equation model of a dynamic system: $\dot{x} = f(x,u,p)$ $y = g(x,p)$ where $x$ is the n-dimensional state vector, $u$ is the r-dimensional input vector, $p$ is ...
0
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1answer
19 views

What the orbits of the following DS look like?

Let us consider the following DS: $$x'(t) = y(t)$$ $$y'(t) = −ax(t) + bx^2(t)$$ I have to describe the orbits of the system when $a>0$ and $b=0$. The eigenvalues of the matrix associated to the ...
0
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1answer
102 views

Integration of the Riccati equation

Simply speaking my goal is to integrate the differential Riccati equation $$x'(t)=a(t)x^2+b(t)x+c(t)$$ I know that this is impossible to express the solution by the means of the elementary functions....
0
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1answer
50 views

Find the equation of the orbit of following ODE

We call orbit associated to the initial datum $(t_0,x_0)$, the set of points: $C=\{$x$(t;t_0,x_0), t \in T\}$. Given the Matrix A: \begin{pmatrix}  0 & 1 \\ -1 & 0 \end{pmatrix} Write the ...
1
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0answers
27 views

Differential algebra in Wikipedia and nlab

The difference between the two definitions is clear in Wikipedia and nlab regarding the definition of a graduated algebra. How to explain this nuance? From Wikipedia: A differential ring is a ...
2
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1answer
134 views

Closed forms of $\int_A^B \sin(\sin(ax))dx$ and $\int_A^B \sin(\sin(ax)) \cdot \sin(\sin(bx))dx$

I would like to know if: $\int_A^B \sin(\sin(ax))dx$ has a closed-form? The solution of Maple requires the presence of Struve functions in its expression. But at least Maple is able to solve it, so ...
4
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1answer
54 views

Are all unital endormorphisms of a Weyl algebra automorphisms?

Given $k$ a field of characteristic $0$, let $A_n(k)$ be the $n$-th Weyl algebra over $k$ – i.e. the unital algebra generated by elements $p_1, ..., p_n$ and $q_1, ..., q_n$ modulo the relations $[p_i,...
0
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1answer
51 views

Accessible first book/resource on differential algebra

I am very interested in learning about the conditions under which a function can be integrated in elementary terms, a topic that I understand falls within the purview of differential algebra. Having ...
8
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4answers
2k views

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, x,...
1
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1answer
45 views

Solved inverse Galois problem for $\mathbb{C}(z)$ seems to contradict the theory about Liouvillian extensions.

The theory about Liouvillian extensions tells us that a Picard-Vessiot extension $L \supset k$ is Liouvillian if and only if the identity component $G^°$ of $G = Gal(L / k)$ is solvable. I think I ...
7
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0answers
196 views

Intuition of Liouville's Theorem (differential algebra) Proof

At the end of my abstract algebra class this spring, we were given an overview of differential algebra and some differential Galois theory. We went too fast to prove anything nontrivial, but I found ...
3
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0answers
38 views

Fixed differential field by a subgroup of a differential Galois group

I am currently reading the book "Differential Galois Theory" by Springer and Van der Put. In the process of establishing that the field fixed by the differential galois group of a picard vessiot ...
1
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0answers
26 views

Need help with the proposition 4.2 in R. C. Churchill's paper on Liouville's Theorem in differential algebra

I'm currently studying Liouville's theorem in differential algebra from this paper. I'm stuck on the proof of the propositions 4.2. In the chapter 4, this "logarithmic derivative identity" is ...
2
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0answers
74 views

Which kinds of equations of elementary functions can have elementary solutions?

Which kinds of zeroing equations of elementary functions can have solutions which are elementary numbers or explicit elementary numbers? Take the equation $$F(x)=0,$$ wherein $F$ is an elementary ...
0
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0answers
51 views

Do these differential operators commute?

Let $D$ denote the differentiation operator on the real functions in $C^1$. For $\alpha\in \Bbb{R}$ and $f\in C^1$ define ${1\over D-\alpha}$ as $${1\over D-\alpha} \,f(t)=e^{\alpha t}\Bigl( \int_0^tf(...
0
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0answers
24 views

German for “Liouvillian extension”

How do I correctly translate "Liouvillian extension" to german, especially "Liouvillian"? "Liouvillsche Erweiterung" sounds rather strange, but might be correct. Anyone knows if this is correct?
0
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2answers
36 views

Finding the solution to differential equation dilemma.

Suppose $\ddot{x_1} =\frac{k(x_2-2x_1)}{3m}$ and $\ddot{x_2}=\frac{k(x_1-x_2)}{2m}$ now how do I solve for $x_1 $ and $x_2$ where $x_2 $& $x_1$ aren't independent of each other ie can't be held ...
0
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1answer
82 views

Cancelling differential terms

Can we cancel two differential terms while they are in a ratio. For example if we have (dx/dt) / (dy/dt), can we just directly cancel dt by dt and write it as dx/dy. I mean is is this step allowed?
20
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1answer
890 views

Why isn't differential Galois theory widely used?

Ellis Kolchin developed differential Galois theory in the 1950s. It seems to be a powerful tool that can decide the solvability and the form of the solutions to a given differential equation. Why isn'...
0
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1answer
75 views

Mistake (?) in differential Galois theory

I have found the following exercise in the book of Crespo and Hajto “Algebraic groups and differential Galois theory”: Let $$\mathcal{L}(Y):=Y^{(n)}+a_{n-1}Y^{(n-1)}+\dots+a_1Y’+a_0=0$$ and let $W$ ...
1
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1answer
49 views

Which kinds of compositions of invertible elementary and nonelementary functions are elementary?

Let $f$ be a bijective elementary function, elementary invertible or not. Let $h$ be a bijective nonelementary function, elementary invertible or not. Which of the compositions $h(f(x))$ and $f(h(x))$ ...
1
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1answer
130 views

Is there any standard nomenclature for the sets of rational, algebraic, and elementary functions?

The rational functions of $X$ can be denoted $\mathbb{C}(X)$, i.e., quotients of polynomials. Is there a standard notation for the algebraic and elementary functions? By the set of elementary ...
2
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0answers
31 views

Showing that a differential Ideal is prime

Consider $\mathbb{R}$ with trivial derivation, $\mathbb{R}\{x\}$ the ring of differential polynomials in $x,$ and let $J$ be the differential ideal generated by $x''+4x.$ In the quotient $\mathbb{R}\{...
0
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1answer
34 views

Proof that the universal first order calculus satisfies its' universal property in the noncommutative case.

Recently I've been reading this paper, and in the first section on (differential calculus on associative algebras) they reference a theorem from Bourbaki's Algebra I Chapter 3. I have a problem with ...
2
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1answer
27 views

Problem with a proof where algebraic extensions are assumed to be finite extensions

I'm reading the article "Integration in Finite Terms" by Maxwell Rosenlicht and I have a problem with one step in a proof. Rosenlicht wants to prove the following: If $F$ is a differential field of ...
1
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0answers
42 views

Creating new constants in differential field extensions via superfluous solutions to a D.E.

I am reading the lecture series book Lectures on Differential Galois Theory by Andy Magid. I came across an example, and I am wondering if it generalizes. The example Let $F_1=\mathbb{C}(z)$ be ...
1
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1answer
54 views

How to use differentiation to show that a curve is symmetric above x axis

If $x^2+y^2=9$, how can I show that this curve is symmetric above $x$-axis using differentiation. Am I correct? Clearly $\frac{dy}{dx} =-\frac{x}{y}$ . So for all $x$ in the domain, $y=+$ or $-$. ...
5
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0answers
79 views

Linear Approximations from Differential Algebras

Suppose we have a differential ring $(R,+,\cdot)$ with derivation $\partial: R\to R$ which is linear $$\partial(f+g)=\partial f+\partial g$$ and obeys the Leibniz rule: $$\partial(f\cdot g)=(\...
5
votes
1answer
1k views

Are all solutions to an ordinary differential equation continuous solutions to the corresponding implied differential equation and vice versa? [duplicate]

Regarding the duplicate. Yes, I know the other one has a lot of shared text, but those were just definitions/setup and I was being lazy. The core questions are still different unless you believe ...
0
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1answer
33 views

Using $\text{SL}(2)$ to solve $2\times2$ linear system of ODEs of order 1 with variable coefficients

Define the matrix $A(t)=\begin{pmatrix} - p(t) & 1 \\ -1 & p(t) \end{pmatrix}$ where $t$ is a real variable and $p$ a function of $t$. Looking for solutions $u=(u_1(t),u_2(t))$ to the ...
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0answers
50 views

How to obtain Appell's PDEs from GKZ-equations?

In F. Beukers' Notes on A-Hypergeometric functions on p. 17, the Appell hypergeometric PDEs are derived from the GKZ-Equations \begin{align*} \partial_{1}\partial_{2}\Phi-\partial_{4}\partial_{5}\Phi &...
0
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0answers
14 views

Use Liouville-Ostrowsi theorem

I don't know how to answer this questions : 1) $e^x/x, exp(exp(x))$ as no elementary primitive. 2) $1/(1+x^2)$ as no elementary primitive in $R(x)$ but have one in $C(x)$. If someone could help me i ...
0
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1answer
22 views

Differentiation, logarithmic fucntion [closed]

The Derivative of $\log_{10} x$ with respect to $x^2$ is? The Answer is having loge(base10)
2
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1answer
629 views

Tangent vectors as derivations

We have such a definition: Given an algebra $A$ over the field $K$, and $M$ is an $A$-bimodule. Derivation is a linear map: $$D:A \to M$$ That satisfies: $D(a*b)=D(a)\bullet b + a\bullet D(b)$ (...
3
votes
3answers
155 views

Constants in localizations of a differential ring

Fix a commutative ring $A$, a derivation $\partial_A$ on $A$, and a multiplicative set $S$ in $A$. Then let $B = S^{-1}A$, $f : A \to B$ be the natural map, and $\partial_B$ be the natural extension ...
15
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1answer
767 views

Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
9
votes
3answers
1k views

Inconsistencies in the definition of derivative of a polynomial over a field

A problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n ...
1
vote
1answer
117 views

Constants in localized differential rings

Fix a differential commutative ring $(A, d_A)$, and a multiplicative set $S$ in $A$. Suppose that $C_A = \{a \in A \mid d_A(a) = 0\}$. Let $B = S^{-1}A$ and transfer the derivation $d_A$ to $B$, ...
0
votes
3answers
121 views

Please solve the differential equation in its simplest form.

$$sec^2(y) \frac{dy}{dx} + 2x \tan(y) = x^3$$ I am not able to get rid of that $dy/dx$ . Please help. PS: I need the answer as $\tan(y) =\frac{x^2 - 1}{2}$
5
votes
1answer
131 views

Is the Antiderivative of an Elementary Function Being Nonelementary Generic?

Many elementary functions, like $e^{-x^2}$ and $\frac{\sin(x)}{x}$ have antiderivatives that are are nonelementary; is this property generic? That is, does the set of all elementary functions whose ...
4
votes
1answer
445 views

Existence of homotopy operator is equivalent to zero homology

Let $(E,\partial)$ be a differential space. A linear map $h:E\to E$ is a homotopy operator if $h\partial+\partial h = {\rm Id}$. Then, there is a homotopy operator in $E$ if and only if $H(E)=0$. ...
2
votes
2answers
410 views

Proof of nonexistence of a solution to an equation in terms of elementary functions?

In a numerical methods class I'm taking, it was claimed that the equation $A = \frac{R^2}{2} \left(\theta - \sin\theta \right)$ cannot be analytically solved for $\theta$. I don't doubt that this is ...