# Questions tagged [differential]

For question about the differential of a map from an open set of a vector space to a vector space.

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### Integrate $\frac{dR}{dt}=\frac{t^3+R^3}{t^2R+tR^2}$ and find the solution for $R(1) = -1$

Yesterday I asked for some help regarding this question which I found from here but I'm stuck again and I'll show you my working so far with my homework First I got this equation separated ...
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### Variable Separable Form for $\frac{dR}{dt}=\frac{t^3+r^3}{t^2r+tr^2}$

I just need a hint or a method on how to separate the equation $$\frac{dR}{dt}=\frac{t^3+R^3}{t^2R+tR^2}$$ so I can integrate it and get a general solution in terms of $R(t)$
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### Derivate by parameter or by initial conditions

Denote in $y(x, τ, ξ)$ solution to the Cauchy problem: $$y' = −y + (x − 3)(1 − e^{−y}), y(τ) = ξ$$ Find: $$\frac{∂y(x, 2, ξ)}{∂ξ}|ξ=0$$ How to do it? Any notes?
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### Differential of the sine function

I know this is very basic, but it's a doubt I have since a couple years. What is the differential of a function like $w = q_1 \sin(\frac{x}{L})$ ? Is it the following? \begin{equation} \delta w = ...
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### What's the differential of the cross product wrt a vector?

$$w= u(x) \times v(x), \qquad u(x), v(x), x \in R^3$$ How to calculate the $\frac{\partial w}{\partial x}$? Thanks very much!
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### Solution of first order linear Differential equations (solving integrating factors) [closed]

An integrating factor of the differential equation - $$(y+\frac{y^3}{3}+\frac{x^2}{2})dx+\frac{x}{4}(1+y^2)dy=0$$
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### Regarding equivalent definitions of Euclidean Submanifolds in Gallot, Hulin and Lafontaine's book Riemannian Geometry

In the book Riemannian Geometry by Gallot, Hulin and Lafontaine, a proposition which characterises equivalent definitions of submanifolds is given as follows: 1.3 Proposition The following are ...
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### Differential Equation, Mice in corner of a square problem [closed]

I'm supposed to solve the mice in the corners of a square or triangle problem, where in a triangle, in every corner is a mouse, and from t = 0s, they start moving towards each other. It is a ...
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### vertical bundle [closed]

Why the vertical bundle equals to ker(dπ)? such that π is the projection space from the bundle E to the manifold M for the fibre bundle. In another book, this is a definition of a vertical bundle. but ...
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### Stability of the origin of the linear system $\dot{x} = Ax$ given $A^2 = I$

Suppose that $A$ is an $n\times n$ matrix such that $A^2 = I$. What can you say about the stability of the origin of the linear system $\dot{x} = Ax$? Are there nontrivial stable and/or unstable ...
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### What is natural way to local trivialize TM (tangent bundle) via normal coordinate?

Suppose dimension n=2 amd M be a Riemannian manifold. We have normal coordinate around p in small ball. In my opinion there are two way to trivialize. Let $(x_1,x_2)$ be a normal coordinate such ...
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### Flows are Stable Under Diffeomorphism

In chapter 2 of this book (entitled: The Simplicity Of Diffeomorphism Groups) the author says that given any compactly supported smooth vector field $V$ on a simply connected and connected (finite-...
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### How do I use an indicial equation to show solutions of ODE? (Frobenius Method)

I'm given this equation:$$xy''+(1-x)y'+\alpha y=0$$, within the indicial equation $$r(r-1)+b_0 r+ c_0 =0$$ I don't understand how should I use the indicial equation to show the two solutions of this ...
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### Nonlinear Differential Equation of High Degree

Any help, please? How can I start to solve them? I tried to use $y'=p$. Also I tried $x=e^x$ and so many methods, but I couldn't reach them to the end. I always got blocked in the middle. Thanks in ...
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### Is d(sinx) and sin(dx) same?

Is d(sinx) and sin(dx) same? If it's not then is there a way to represent sin(dx) or for that matter any function f(x) where x is put to be some differential? Examples are most welcome if needed.
If I have a function of two variables, say $f=f(x,y)$, I know that its differential is computed as $$df=\frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dx$$ But is it possible to ...