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Does the following make logical mathematical sense: $$x^2=t$$ $$\frac{d} {dy} (x^2)=\frac{d} {dy} (t)$$ $$2x\cdot\frac{dx}{dy}=\frac{dt} {dy}$$ $\mathbf{\therefore \frac{dy} {dx} =2x \cdot\frac{dy}{... 1answer 81 views ### Reduction modulo a prime ideal I am looking at the following part of a paper:   When we reduce the differential equation$(1)$modulo the prime$p$we do the following: $$\alpha_i \equiv \tilde{\alpha}_i \... 1answer 128 views ### May the integral \int\root 3 \of{\cos(x)^2}\,dx be expressible by elementary functions? I would like to decide by methods of differential algebra whether the integral \int\root 3 \of{\cos(x)^2}\,dx might be contrary to the output of CAS Mathematica Online Integrator expressible by ... 0answers 286 views ### Why differential Galois theory is not widely used? E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation. My ... 0answers 124 views ### How does commutative and/or differential algebra think about total derivatives? If we apply the "operator" \frac{d}{dx} to the polynomial xy, we get the expression y+x\frac{dy}{dx}. (Source: high school.) Thinking of xy as an element of the polynomial ring \mathbb{R}[x,y]... 0answers 281 views ### chain rule for derivations Off we go. So let b:X\rightarrow Y be a function from X to Y endowed with as much structure as it needs to make sense of the question :) and a:Y\rightarrow \mathbb R a function into the reals. ... 0answers 44 views ### If D:A\to A is a derivation, what can be said about the range of D? What can be said about the relation between the domain and range of a derivation as a function? If A is the domain, any space of functions, what does D(A) look like, where D is a derivation? ... 0answers 30 views ### The spectral transfinite open spaces with quintic characteristics of second kind Context: Beginning with the formal definition of transfinite spaces together with the Picker-Hansel theorem, we obviously get a relation$$ \bigcap\xi_{|\sigma|\mapsto \theta^*} \oplus_\psi \left(\... 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, Picardâ€“... 0answers 29 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 ... 0answers 33 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 + (...
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 \$...