# Questions tagged [differential-algebra]

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

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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/...
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+...
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 $(\mathfrak{E},Y)$...
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### 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 ...
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 ...
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 $\delta:R\to R$ ...
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 $\mathfrak{g}$-...
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### 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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### 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 ...
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 ...
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....
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### 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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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### 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 ...
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 ...
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 &...
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### 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 ...
22 views

### Differentiation, logarithmic fucntion [closed]

The Derivative of $\log_{10} x$ with respect to $x^2$ is? The Answer is having loge(base10)
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: $D(a*b)=D(a)\bullet b + a\bullet D(b)$ (...
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 ...