# Tagged Questions

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### Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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### Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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### Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
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### Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Eulerâ€“Lagrange ...
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### Elliptic partial differential equations

Consider the following elliptic PDE: $$\Delta u=f(u),$$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
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### how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
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### Calculus of Variations

In the Calculus of Variations there is a passage from Euler's characteristic equation: $$\frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0$$ in ...
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### Calculus of variation: Reduce the order of a differential equation using a 1 parameter lie group adfmitted by it.

We are asked to reduce $y^"+y-y^{-3}=0$ using $X= \sin2x\frac{\partial}{\partial x}+y\cos2x\frac{\partial}{\partial y}$ I know we have to find the first prolongation of X and solve $X^1$F=0 using ...
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### Euler-Lagrange Calculus of Variations Example

I have been working on solving Euler-Lagrange Equation problems in attempts to learn Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration. I am ...
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### Maximizing an integral through maximum principle

Suppose that we wish to achieve $$\max\int_0^1 (1-x^2-\dot{x}^2)dt, x(0)=0, x(1)\geq 1$$ Two possible ways one can do this is by Euler-Lagrange eqn or maximum principle. Applying the Euler-Lagrange ...
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### Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
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### stationary function of an integral

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$. I don't know what a stationary function is. Can you anyone suggest me ...
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### Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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### Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
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### Differential Equations for a Teardrop Shape

My research has led me to a nonlinear system of differential equations which should yield a teardrop shape in the $x-y$ plane. The equations, parameterized by $t$ are ...
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### Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
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### Euler-Lagrange problem solution

Hi, Can anyone solve this question? I have no clue.
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### Euler lagrange equation solving

Find the Euler-Lagrange equation for the functional $$I(y) = \int_0^1(py\,'\,^2-qy^2)\mathrm dx$$ subject to the constraint $$\int_0^1ry^2 = 1.$$ Answer: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...