# Tagged Questions

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### Finding function given its Jacobian and the initial condition

Consider continuously differentiable function $f:\mathbb{R}^k\mapsto \mathbb{R}^k$. We know that $f(x_0)=y_0$ and the Jacobian matrix is given for all $x$. I'd like to know the explicit for of the ...
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### $\textbf c(t)$ is a flow line on $\textbf F = -\nabla V$, prove $V(\textbf c(t))$ is a decreasing function of $t$.

Let $\textbf c(t)$ be a flow line of a gradient field $\textbf F = -\nabla V$. Prove that $V(\textbf c(t))$ is a decreasing function of $t$. We have not learned Line Integrals, so I would assume this ...
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### how to solve this homogeneous differention equation ?

$dy/dx= (2x+3y+4)/(4x+6y+5)$. I am trying to solve this homogeneous Ds, but don't understand how to solve it. I believe the first step is to solve this: 1) $y=u x$ 2) $dy/dx = u+x \cdot du/dx$ then ...
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### How to interpret positive eigenvector of $A$ in $\frac{dx}{dt}=Ax$?

I was doing some simulations of diff. equations in Matlab. I had a matrix $A$ with all negative eigenvalues. Also the eigenvector corresponding to the greatest eigenvalue had all positive elements. I ...
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### A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
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### derivation of higher order numerical methods for ODEs using Mathematica, Matlab, Maple

I want to know that is it possible to expand the multi-variable Taylor series of f(x+ah,y+ bhf(x+ch,y+dh)) in Mathematica. My purpose is to construct higher order methods which is very typical to ...
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### Multivariable differential equation

Given $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure where ...
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### Integrating two equations that equal, what happens to the constant on one of the sides?

In class, we were talking about Newton's 3rd law and how to integrate. $\int(g)dt = \int(y''(t))dt \implies g(t) + C = y'(t)$ I am confused about why the right hand side of the equation doesn't get ...
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### Laplace's Equation with Neumann BC

Hi fellow math enthusiasts, I am currently working on some research to do with the electric field induced within the brain via magnetic stimulation. I am trying to solve the partial differential ...
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### Help with solving for a flow curve:

So I'm preparing for a final exam in multivariable and our textbook posed the following question: find the flow lines of F(x,y) = (-y, x) Which I can't seem to solve correctly. We are told that a ...
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### Motion in three dimensions with friction.

I am trying to represent the motion of an object in three-dimensional space that is undergoing acceleration, friction, and drag, where: acceleration = $\vec A$ friction = $F$ drag = $D$ The ...
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### How to find other solutions to this vectorproblem?

Suppose I have a vector field $\mathbf{A}(x,y,z)$, of which I know: $$\mathbf{A}(x,y,0)=(1+\alpha x)\hat{z}$$ Thus, I know the value of $\mathbf{A}$ in the $xy$-plane. Say, within ...
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### analytical solution for system of nonlinear ODEs in two dimensions

This ODE comes out of a system of self propelled particles (http://www.foelsche.com/swarm). Currently this system is linear -- I want to add a nonlinear friction/drag. Every particle has the following ...
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### $o(|X(t+\Delta t)-X(t)|)$ is $o(\Delta t)$?

X is differentiable. if $X'(t) \neq 0$ , it's easy to show. $\lim_{\Delta t \to 0} \frac{o(|X(t+\Delta t)-X(t)|)}{|X(t+\Delta t)-X(t)|} \frac{|X(t+\Delta t)-X(t)|}{\Delta t} = 0 \cdot |X'(t)|$ ...
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### arbitrary patch in terms of orthogonal patch

Let $\mathcal{M}$ be a $2$ dimensional differentiable manifold (a geometric surface). Let $\mathbf{y}$ be an arbitrary patch in $\mathcal{M}$. How can one prove that $y$ can be expressed as ...
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### Finding the characteristic ODE from a nonlinear PDE

I am studying for a PDE exam on Tuesday, and I am getting pretty confused about one specific type of problem and I am thinking that perhaps I am misinterpreting the correct procedure to follow. The ...
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### why $\frac{\partial f(x,y)}{\partial x}+f(x,y)\frac{\partial f(x,y)}{\partial y}=0 \Rightarrow f(x,y) \equiv \text{constant}$

Assume $f(x,y) \in C^{(1)}(\Bbb{R}^2)$, if$$\frac{\partial f(x,y)}{\partial x}+f(x,y)\frac{\partial f(x,y)}{\partial y}=0$$. Show that $f(x,y) \equiv \text{constant}$ My approach: For every solution ...
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### If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
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### 2- norm of a vector in spherical and cylindrical coordinates

I was wondering how the 2-norm of a vector in cylindrical and spherical coordinates looks like? Or more general, what is the idea to derive it? Does anybody have an idea?
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### dropping a particle into a vector field, part 3

Okay, so I've been independently trying to study basic systems of differential equations as they relate to dropping a particle into a vector field. I have had two previous posts on the matter trying ...
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### dropping a particle into a vector field, part 2

Okay, so earlier I posted this question "dropping a particle into a vector field " as sort of a feeler question as i study line integrals in order to go into surface integrals and eventually ...
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### dropping a particle into a vector field

I'm independently studying Colley's Vector Calculus and am on the section on line integrals. I understand that the line integral gives the amount of work done on a vector field for a predetermined ...
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### Prove the following function is decreasing

Given: $E[u(t)] = \int_{\Omega} \left(\frac{1}{2}\left|\nabla u\right|^{2} - \frac{1}{4}u^{4}\right)dx$, and $u_{t} - \nabla u = u^{3}$. Show: $\frac{\partial}{\partial t}E\left[u(t)\right] \le 0$. ...
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### Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
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### Finding the jacobian of a differential system with a piecewise function

My system: $$\frac{\mathrm{dx} }{\mathrm{d} t}=-ax^2+y^2-\gamma z$$ $$\frac{\mathrm{dy} }{\mathrm{d} t}=- h(y)-\beta y$$ $$\frac{\mathrm{dz} }{\mathrm{d} t}=x+h(y)-\beta z$$ where $h$ is the ...
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### Conditions for multivariable differentiability

For a nonlinear system of equations: $x' = f(x)$ why is it that the following condition is equivalent to being differentiable at $x_0$. Condition: f(x) = A(x-x_0) + g(x) ...
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### Decomposite a vector field into two parts

Let A be a region in $\mathbb R^3$, and suppose $\vec {\mathbf F}$ is a smooth vector field on A. I was asked to show that I can write $\vec {\mathbf F}=\vec {\mathbf F_1}+\vec {\mathbf F_2}$, s.t. ...
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### Linear dependence of multivariable functions

It is well known that the Wronskian is a great tool for checking the linear dependence between a set of functions of one variable. Is there a similar way of checking linear dependance between two ...
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### ODE phase portrait and vector function interpretation

I do not quite remember how to plot a vector function (or maybe I do). Consider the ODE: $$x' = \begin{pmatrix}1&1\\-1&1\end{pmatrix}x$$ I have found the general ...
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### Solve the following differential equations by converting to Clairaut's form through suitable substitutions.

The question comprises of three subparts which need to be converted to Clairaut's form through suitable substitutions and then solved : (a) x p2 - 2yp + x + 2y = 0 (b) x2 p2 + yp (2x + y) + y2 = 0 ...
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### Solving a 5 dimensional function in a neighbourhood

Consider a function $f:\mathbb{R}^5 \to \mathbb{R}^2$ defined by $$f(u,v,w,x,y)=(uy+vx+w+x^2,uvw+x+y+1)$$ such that $f(2,1,0,-1,0)=(0,0)$ (i) Show that we can solve $f(u,v,w,x,y) = (0,0)$ for ...
How to examine if functions: $f(x,y)=|x+y|$ and $g(x,y)=\sqrt{|xy|}$ are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$