Variation Definition and 574 Threads

First time poster, so please feel free to leave any comments of a general nature. I'm hoping to get a further insight on the derivation of the variation of parameters method used in ordinary differential equations to solve linear second order equations. I understand were looking for a...

Homework Statement A thin ring of radius x carries a charge Q uniformly distributed around it. Sketch garphs showing how the potential and the field vary along the axis (y) from its centre. A small dipole of strength p is placed at the centre of the ring so as to point alonf th axix (going...

Hey all, this is a little confusing, because the "variation of parameters" that I have been taught in class is different then what I find in most texts... I have y''' + y' = tan(x) Most textbooks use the wronskian and work from there, what I was taught to do is set it up as the...

This is what a book says - “The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of the electric field in that region. Thus, the field lines are close together where the electric field is strong and far apart where...

Homework Statement http://img27.imageshack.us/img27/6083/variationofparametersfop.jpg

I am trying to find a clear answer as to why elements' weights vary, when their makeup are of the same protons, electrons, and neutrons? For example, H = 1.0079 atomic weight, 1 proton + 1 electron Li = 6.941 atomic weight, 3 protons + 3 electrons + 4 neutrons If a neutron = 1 proton + 1...

how does cosmology explain that the creation of the universe was not a perfect uniform event in all directions type thing. I mean if you took a chunk of the universe and looked at it the density of matter wouldn't be uniform some planets are bigger than others and so forth. Maybe I am looking at...

Homework Statement Find the general solution of the following diff. eqn. y''(t) + 4y'(t) + 4y(t) = t^(-2)*e^(-2t) where t>0 Homework Equations General soln - Φgeneral(t) + Φparticular(t) Wronskian - Φ1(t)Φ22'(t) - Φ2(t)Φ1'(t) The Attempt at a Solution I'm solving by...

Homework Statement y''+y=tan(x)+e^{3x}-1 Homework Equations homogeneous solution: y_{hom..}=C_{1}cos(x)+C_{2}sin(x) particular solution: y_{parti..}=v_{1}' cos(x)+v_{2}' sin(x) The Attempt at a Solution v_{1}' cos(x)+v_{2}' sin(x)=0 (1) -v_{1}' sin(x)+v_{2}' cos(x) =...

Homework Statement Find the general solution using the method of variation of parameters of: y''-6y'+9y=(x^-3)(e^3x) I found the roots of the corresponding homogeneous equation to be lamba = 3. So there are repeated roots. My question is, how do I solve a variation of parameter...

Homework Statement A system that consists of n moles of ideal gas does a free expansion (to the vacuum) from a volume V to a volume 2V. a) What is the variation of entropy of the gas?; b) of the universe?; c) if the expansion was reversible and isothermal, what would be the variation of the...

Homework Statement A metal wire has a resistance of 8.10 at a temperature of 20°C. If the same wire has a resistance of 11.45 at 90°C, what is the resistance of the wire when its temperature is -20°C? Homework Equations R=R(o)[1+alpha(T-To)] alpha=R-Ro/Ro(T-To) The Attempt at a...

Hello all :) Homework Statement I'm trying to understand the fundamentals of General Relativity, but alas, I seem to be unable to grasp the fundamentals of variational calculus. Specifically, I'd like to prove the following relation for the square root of the negated determinant of the...

Homework Statement Show that variation principle (parameters ci) leads to equations \sum\limits_{i = 1}^n {\left\langle i \right|H\left| j \right\rangle c_j = Ec_i {\rm{ where }}} \left\langle j \right|H\left| i \right\rangle = \int {d\textbf{r}^3 \chi _j^* \left( \textbf{r} \right)\left(...

I am looking for information on the density of a liquid at lower temperatures. I have it at 15C as 725 kg/m^3. The fluid is avgas 100LL. I wish to determine it at lower temperatures. I believe that the change in liquid densitiess is rather small for temperature changes, but I can't verify...

Hi, When using the method of variation of parameters to solve something like; y'' + y' = 2^x I got the aux. equation: r^2 - r =0 which gives the roots r=0,1 How do I find the complementary equation yc?

The Einstein field equations \mathsf{G} = \kappa \mathsf{T} can be derived by considering stationary metric variations of the Einstein Hilbert action, S = \int \mathrm{d}^4x \sqrt{-g} (R/2\kappa + \mathcal{L}_\mathrm{M}). 0 = \delta S = \int\mathrm{d}^4...

Consider, x' = x + 3y^3 y' = -3y I am trying to use the fundamental matrix, F(t), and 3y^3 as my g(t) in order to plug into the variation of parameters formula... Xp = F(t) * \integral{ F(t)^-1 * g(t) } , Am I going about this the wrong way? I am trying to get...

Hi I know that our world is quite unstable and minor changes in these parameters could make the existence of life impossible. However, I am interested in what exactly is going to happen if we start to increase/decrease any of 30 parameters. It is interesting how well-tuned these parameters...

Homework Statement What is the unit of the system-universe temperature variation coefficient? The system is a container holding a mass of water. T The universe I guess is the room temperature. Troom t is time in seconds. \Psi is the temperature variation coefficient. Homework Equations...

What is "Variation of Parameters"? Homework Statement None. General. Homework Equations I don't know. :( ? The Attempt at a Solution ? I am taking a class right now on engineering analysis (which I am finding it to be more like partial differential equations mixed with...

Homework Statement y''+2y'+y = 4t^2 - 3 + (e^-t)/t of course i evaluated the general soltuion to be c1e^-1t + c2te^-1t but now how do you do the right part? i tried y=At^2+Bt+c+1/(Dt+E)*e^-t as a solution but after differentiating it twice and putting it into the eqaution i got...

Question is attached as Clipboard01.jpg I have tried the use Variation of Parameters to solve this question, but I kept getting wrong answer. This is What I get y=(2e^x)(Cos(e^x))+0.5(e^(-x))Cos(e^(-x))-2Sin(e^(-x)) This is the right answer: y=-Sin(e^(-x))-(e^x)Cos(e^(-x)) Procedure is...

I was reading about the principle of least action and how to derive Newton's second out of it. at a certain point I didn't follow the calculations, so the author defines a variation in the path, x(t) \longrightarrow x'(t) = x(t) + a(t), a \ll x a(t_1) = a(t_2) = 0 Now, S...

As the name suggest, this problem is an undetermined coefficients problems where variation of parameters is necessary to solve. As with my previous question; This is not a homework problem, but it is out of the textbook so I figured this would be the appropriate place to ask if I am doing it...

In the Nambu-Goto action, the Lagrange Density is L(dX/dt,dX/dx). When the action is varied, this term becomes a sum of conjugate momentums multiplied by their respective instantaneous velocities, i.e. (Px*dX/dx + Pt*dX/dt). Nowhere, can I find an detailed explanation of how one gets from the...

I recently got this as an idea for an interesting independent project. I had planned to do the experiment using the four point method of resistance measurement, but I'm stuck on figuring out just what materials I might need to use. I intend to use the basic high school equipment, nothing too...

What role do you think variations play in the process of evolution?

So the problem I am working on is.. A 100 cm long copper wire of radius 0.45 cm has a potential difference across it sufficient to produce a current of 5.0 A at 20°C. Find a) What is the potential difference. b) If the temperature of the wire is increased to 200°C, what potential difference is...

The two dimensional action is: S_k = \int d^2\sigma\sqrt{h}\left(\partial_\alpha\phi\partial^\alpha\phi - \frac{i}{2}kR^{(2)}\phi\right) where k is a constant, R^{(2)} is the two dimensional scalar curvature. I'm trying to derive the following energy momentum tensor: T_{\alpha\beta}^k =...

Homework Statement Let y(x) represent the path of light through a variable transparent medium. The speed of light at some point (x,y) in the medium is a function of x alone and is written c(x). Write down an expression for the time T taken for the light to travel along some arbitrary path y(x)...

Homework Statement This is the problem 8.10 from Levine's Quantum Chemistry 5th edition: Prove that, for a system with nondegenerate ground state, \int \phi^{*} \hat{H} \phi d\tau>E_{1}, if \phi is any normalized, well-behaved function that is not equal to the true ground-state wave function...

As I understand it, a tesla turbine runs by injecting steam, or some other gas onto the outside of several disks mounted parallel to each other on a shaft. The gas is directed roughly tangent to the outside of the disks and it exits through openings in the center of the disks. Seems straight...

EDIT: Sorry... I have to use perturbation theory. My mistake. Hey... I have a quick question. I have to calculate the approximate change in energy via variation theory when the 'error' Hamiltonian for the Stark effect is defined as: |\vec{E}|cos\theta\bullet eR If I'm not mistaken, the change...

After the "I DERIVE YOU" math joke (posted somewhere in the stickys) Here is one having a similar punchline, but, in my son's opinion, with a MUCH better joke: A constant function and an exponential function are walking down the street and they see a partial derivative. "Go over and talk to...

Sorry about twin paradox again, I'll try to keep this simple as simple as I can. Initial situation: All participants are at rest, A1 and A2 are together at midpoint between B1 and B2. All clocks are in sync. _____________A1, A2 _____B1____________________B2 ---- A1 and A2 move...

Homework Statement Give examples of methods used to find approx. answers to the Schrödinger's eqn. of systems too complex to be solved analytically. Homework Equations None required. The Attempt at a Solution I understand there are three methods commonly used, those mentioned in...

[SOLVED] Variation of Parameters Homework Statement y^(4)-6y^(3)=-5sinx The Attempt at a Solution I factored this at x^3(x-6)=0 so my r values are 0,6 also using for y(p) Dcosx + Esinx y=Ae^0 + Be^6x + Dcosx + Esinx ? y' =6Be^6x -Dsinx + Ecosx y'' =36Be^6x-Dcosx - Esinx...

Homework Statement Okay so when solving a system of D.E.s using Variation of Parameters I know that first I find the complementary solution Xc and then do a bunch a of crap after that using the fundamental matrix. Now I just came across a problem with repeated roots, so I just want to...

So I pretty much have this Differential Equation solved except that I have to integrate the expression \int \Phi(t)F(t)dt it has a star next to it in my attached work. Does this look readily integrable to anyone? For some reason nothing is ringing a bell. I suppose I could go by parts, but...

Recall that some traits, such as skin color, appear to be controlled by several genes. This creates a continuum of variation. If this polygenic explanation for the inheritance of human skin pigmentation is correct, how do the skin colors of the following 4 individuals compare? Which of the...

Hi, Here's the problem: Homework Statement Quantum particle moving in 1D. Potential energy function is V(x) = C|x|^{3}. Using the variational method, find an approx. ground-state wave function for the particle. The Attempt at a Solution Using \psi = Ae^{-ax^{2}}, I find that A =...

FTL Signaling with Dopfer - Why Won't This Work? I think at the outset we will all agree that the following will not work, but the more interesting question will be why. Take the Dopfer experiment described at: http://www.quantum.univie.ac.at/publications/thesis/bddiss.pdf On page 36...

Solve by method of variation of parameters (x^2)y'' - (4x)y' + 6y = x^4*sinx (x > 0) Hey, I know how to solve problems using variation of parameters but only when the corresponding homogenous equation has constant coefficients... y'' - (4/x)y' + (6/x^2)y = 0.. the bit I am confused about...

I read on NIST CODATA that electron mass is a fundamental constant. I take from this that proton mass is not fundamental. I also read it is possible that proton-electron mass ratio may vary. So I wonder how proton mass is measured or derived. I have no training in physics, however in studying a...

Homework Statement Use the method of variation of parameters to determine the general solution of the given differential equation: y^(4) + 2y'' + y = sin(t) Homework Equations characteristic equation is factored down to (r^2 + 1)^2, so r = +/- i. this gives the general solution to be...

Homework Statement Suppose a function Aexp([-cr]^{2}[/tex]. To find optimized value of c, using variational method. Relevant equations[/b] The Attempt at a Solution I even don't understand how to start doing it. Is their anybody who can help me?

"What your favorite variation of" : Euler's Formula I generally find that mathematicians always have a preferred way of writing an expression, whether be it because to them it's more aesthetic pleasing or easier to memorize. Few expressions, however, lend themselves to many forms as thus...

Within the description for the variation of parameters procedure is the restriction: y1u1' + y2u2' = 0. Can you explain this restriction, it is not obvious to me, I do not have an explanation where this comes from. Is it related to u[ \frac {dy}{dx} + P(x)y] = 0 from solving first...

Greetings, Regarding the two procedures: undetermined coefficients and variation of parameters, can both procedures be used interchangeably - meaning they both solve (non-homogeneous linear equations)? Does one method work better in certain situations, if so which method is preferred when...