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0
votes
2answers
39 views

Differentiating Query

Does the following make logical mathematical sense: $$x^2=t$$ $$\frac{d} {dy} (x^2)=\frac{d} {dy} (t)$$ $$2x\cdot\frac{dx}{dy}=\frac{dt} {dy} $$ $\mathbf{\therefore \frac{dy} {dx} =2x \cdot\frac{dy}{...
2
votes
1answer
79 views

Reduction modulo a prime ideal

I am looking at the following part of a paper: $$$$ $$$$ When we reduce the differential equation $(1)$ modulo the prime $p$ we do the following: $$\alpha_i \equiv \tilde{\alpha}_i \...
1
vote
1answer
128 views

May the integral $\int\root 3 \of{\cos(x)^2}\,dx$ be expressible by elementary functions?

I would like to decide by methods of differential algebra whether the integral $\int\root 3 \of{\cos(x)^2}\,dx$ might be contrary to the output of CAS Mathematica Online Integrator expressible by ...
10
votes
0answers
276 views

Why differential Galois theory is 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. My ...
5
votes
0answers
124 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring $\mathbb{R}[x,y]...
5
votes
0answers
277 views

chain rule for derivations

Off we go. So let $b:X\rightarrow Y$ be a function from $X$ to $Y$ endowed with as much structure as it needs to make sense of the question :) and $a:Y\rightarrow \mathbb R$ a function into the reals. ...
4
votes
0answers
44 views

If $D:A\to A$ is a derivation, what can be said about the range of $D$?

What can be said about the relation between the domain and range of a derivation as a function? If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? ...
3
votes
0answers
30 views

The spectral transfinite open spaces with quintic characteristics of second kind

Context: Beginning with the formal definition of transfinite spaces together with the Picker-Hansel theorem, we obviously get a relation $$ \bigcap\xi_{|\sigma|\mapsto \theta^*} \oplus_\psi \left(\...
2
votes
0answers
167 views

What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, Picard–...
1
vote
0answers
26 views

Differential operator and multi-index

By induction it can prove Leibnitz rules $\displaystyle D^\alpha(fg)=\sum_{|\beta| \leq |\alpha|} \binom{\alpha}{\beta} D^\beta f D^{\alpha - \beta} g$ from the book where I'm studying, it says that ...
1
vote
0answers
32 views

Index reduction for DAE

I have to simulate a set of DAE's. Therefore I have to reduce the index for this problem: $ (ms+mb)*\ddot z + mb*ls* \ddot \phi s + mb*lg* \ddot \phi b = -(ms+mb)*g - \lambda2$ $ (mb*ls)*\ddot z + (...
1
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0answers
20 views

Adjoining a derivative

Let $u$ be an algebraic solution of $y'=(1/x)(y^2 + y^3)$ other than $-1$ and $0$ over $\Bbb{C}(x)$ (the quotient field of $\Bbb{C}[x]$). So, $u$ is some fractional power series. Suppose, we adjoin $...
1
vote
0answers
40 views

Differentiating both sides of a DE

In general if you have a differential equation with two variables such that: $$L(x,y)=h_1[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ R(x,y)=h_2[f(x),f'(x),f^{(2)}(x),....
1
vote
0answers
79 views

Algebras vs. rings in algebraic differential calculus

Vector fields (and differential operators of higher order) on a real manifold are often defined in terms of $\mathbb{R}$-algebra $C^\infty(M)$. However, it is not clear to me why is the $\mathbb{R}$-...
0
votes
0answers
39 views

Does the “Leibniz multicategory over $R$” have an accepted name?

Let $R$ denote a commutative ring. Definition. The "Leibniz multicategory" over $R$ is given as follows: Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate'). ...
0
votes
0answers
29 views

Solvability of linear homogeneous differential equation in terms of lower order

Let $ L(y)=0$ be a linear homogeneous differential equation with coefficients in a differential field $k$. We say that a differential field $K$ containing $k$ is a Picard-Vessiot extension of $k$ for ...
0
votes
0answers
18 views

Determine a change of variables to transform one DE to another

Given two ODE's, is it possible to determine if one can be obtained from the other via a change of variables? In particular, I have the two ODEs: $$ \begin{split} \frac{d^2y}{dt^2}&=\frac{8t-3}{...
0
votes
0answers
14 views

Finite extension of a closed differential field is closed.

Let G be the differential Galois group of a differential field extension M of K. Then any finite dimensional extension of a closed intermediate differential field is closed. I can not realise this ...