Variation Definition and 574 Threads

Hi everyone, :) One of my friends gave me the following question. I am posting the question and the answer here so that he could check his work. Question: This question concerns the differential equation, \[x\frac{d^{2}y}{dx^2}-(x+1)\frac{dy}{dx}+y=x^2\] and the associated homogeneous...

My teacher said that instead of using a fiber optic made of just one material we could use a set of materials with progressively low refractive index to turn the light back in, like so: In the case of a continuous variation of refraction index, how can you do the math. Can you please show...

It is generally well known that the elastic modulus of most materials become larger with decreasing size. This could be due to decreased number of dislocations, surface effects etc. Does anyone know how exactly does this increase? Considering a ceramic or an oxide, how does the Elastic modulus E...

Homework Statement Prove that if a function f is once-differentiable on the interval [a, b], then Vf = \int ^{b}_{a} | f'(x) | dx, where Vf = sup_{P} \sum^{i=n-1}_{0} | f(x_{i+1}) - f(x_{i}) | where the supremum is taken over all partitions P = \left\{ a = x_{0} < x_{1} < ... < x_{n}...

I learned ΔSuniv = ΔSsys + ΔSneib If ΔSuniv > 0 the process is spontaneous. But ΔS = Q/T right? Shouldn't Qsys = -Qneib? (The heat released by the system = - heat released by the neighborhood) I cannot understand why this is wrong! If it were the case, ΔSuniv would be always zero...

I would like to hear opinions on the variation in decay rates as described by Fischbach and coworkers and how (if at all) this will affect radioemtric dating. Does this phenomenon indeed exist or is it the result of errors in experimental technique?

Not really a homework question, just the notes are confusing me. Homework Statement Let S be a functional. (Given without proof) If S is differentiable its derivative \delta S is uniquely defined as \delta S = \int_{x_{0}}^{x_{1}}\frac{\delta S}{\delta \gamma} \delta \gamma dx where...

Hi friends, going through Palatini gravity, I cannot do the variation for palatini f(R) gravity and get to the famous equation (Tsujikawa dark energy book equation 9.6): R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R =\frac{\kappa^2 T_{\mu\nu}}{F} - \frac{FR-f}{2F}g_{\mu\nu} + \frac{1}{F}(\nabla_\mu...

This is the barn pole paradox using a slightly curved pole and a silo instead of a perfectly straight pole and doors on a barn. Let's say we have a pole of proper length of 1 light-year that is traveling around a circle that has a curvature of 1 meter over 1 light-year of length. Let that pole...

Hi, I am wondering why radial variation in Poisson's ratio exists when a cylinder is compressed? Ie, Poisson's ratio toward the center is lower than what is measured at the circumferential edge. Thanks M

I have come across a MATLAB code for solving the spruce budworm differential equation.But,I would like to solve the same differential equation for a range of parameters(r=0:5,q=0:10).I am having problems trying to define the array of matrices for each loop.Please look at the code below:% This...

Homework Statement This is not homework but more like self-study - thought I'd post it here anyway. I'm taking the variation of the determinant of the metric tensor: \delta(det[g\mu\nu]). Homework Equations The answer is \delta(det[g\mu\nu]) =det[g\mu\nu] g\mu\nu...

I am considering the variation of \delta ( \sqrt{g} R_{abcd} R^{abcd} ) and I know the answer is - \frac12 \sqrt{g} g_{\mu\nu}R_{abcd} R^{abcd} +\sqrt{g} R_{( \mu}{}^{bcd} R_{\nu ) bcd} + \ldots what i do not understand is the coefficient of the last term. For example, when we...

The ODE to solve via variation of parameters is ##(1-x)y''+xy'-y=(1-x)^2##. Knowing that ##e^x## and ##x## are solutions to the homogeneous ODE. Now if I call ##y_1=x## and ##y_2=e^x##, the Wronskian is ##W(y_1,y_2)=e^{x}(x-1)##. According to...

Homework Statement I must solve ##y''+2y'+2y=e^{-t}\sin t##. I know variation of parameters might not be the fastest/better way to solve this problem but I wanted to practice it as I never, ever, could solve a DE with it. (Still can't with this one). Though the method is supposed to work...

Homework Statement At time t < 0 there is an infinite potential for x<0 and for x>0 the potential is 1/2m*w^2*x^2 (harmonic oscillator potential. Then at time t = 0 the potential is 1/2*m*w^2*x^2 for all x. The particle is in the ground state. Assume t = 0+ = 0- a) what is the probability that...

Homework Statement the spring of the pressure gauge. has a force constant of 1250 N/m, and the piston has a diameter of .012m. As the gauge is lowered into water in a lake, what change in depth causes the piston to move by .0075 m Homework Equations P= P_{0} + \rhogh variation with...

x²y"(x)-3xy'(x)+3y(x)=2(x^4)(e^x) =>y"(x)-(3/x)y'(x)+(3/x²)y(x)=2x²e^x i don't know how to approach this problem because the coefficients are not constant and i am used to being given y1 and y2 HELP!

1a. Homework Statement Hi all, I'm having quite a bit of problem on a physics question. I've seen some examples of this problem, but when I follow the suggestions and equations out, the answer I get back is wrong... Anyways, here's the problem: Given two masses (m_1,~m_2) hanging from...

Here's the proof that I read for method of variation of parameters- https://www.physicsforums.com/attachment.php?attachmentid=52267&stc=1&d=1351081780 What I couldn't understand is that how could one simply assume that u'1y1+u2'y2=0 and u'1y'1+u2'y'2=g(x) I just don't understand...

I was recently trying to prove the variation of parameters formula for an nth degree equation, and I have come up with a question about the assumptions made during the derivation. During the derivation we assume that: u1'y1(k) + u2'y2(k) + . . . + un'yn(k) = 0 for k < n-1. It leads to the...

Homework Statement One body of constant pressure heat capacity C_P at temperature T_i it's placed in contact with a thermal reservoir at a higher temperature Tf. Pressure is kept constant until the body achieves equilibrium with the reservoir. a) Show that the variation in the entropy of the...

Homework Statement Hello, I would like your comments on my observations. I have a set of data from a step experiment, one input (volts D.C), one output (temperature). By doing system identification, the resultant model is second order and its poles lie at 0.99 and 0.8 (z-plane) Now for...

I am familiar with basic calculus of variations. For example, how to find a function that makes some integral functional stationary (Euler-Lagrange Equations). Or for example, how to perform that same problem but with some additional holonomic constraint or with some integral constraint. The...

I have a very limited knowledge of tensor calculus, and I've never had proper exposure to general relativity, but I hope that the people reading this forum are able to help out. So I'm doing some stat. mech. and a part of a system's free energy is \mathcal{F} = \int V(\rho)\nabla^2\rho dx I'd...

Homework Statement Folks, how is the following expansion obtained for the following function ##F(x,u,u')## where x is the independent variable. The change ##\epsilon v## in ##u## where ##\epsilon## is a constant and ##v## is a function is called the variation of ##u## and denoted by...

I have the equation x + y = z. Z is a constant. What type of variation is expressed here? An example of direct variation is x*z = y. An example of inverse variation of x * y = z. In both examples, z is a constant. So what's the answer?

Question: Find the total kinetic energy per unit volume in a monoatomic gas at standard temperature and pressure and deduce an expression for the variation of this kinetic energy with temperature if the pressure is maintained constant. [Standard pressure = 1.01E5 Pa] Attempt: Pressure...

Homework Statement y''-2y'+y = \frac{e^x}{1+x^2} Homework Equations u_1 = -\int \frac{y_{2}g(x)}{W}dx u_2 = \int \frac{y_{1}g(x)}{W}dx g(x) = \frac{e^x}{1+x^2} W is the wronskian of y1 and y2. The Attempt at a Solution The characteristic equation for the homogenous solution...

Is it possible to take the variation of the Dirac delta function, by that I mean take the functional derivative of the Dirac delta function?

Homework Statement solve the following differential equation: t4x'' - 4t3t' + 6t2x = - 12t - 20 Homework Equations substitution x(t) = tn The Attempt at a Solution this is a Euler equation with the following general solution: x(t) = c1t2 + c2t3 worked out using the above substitution. The...

The variation method for approximating the the ground state eigenvalue, when applied to higher energy states requires that the trial function be orthogonal to the lower energy eigenfunctions.In that respect this book I am referring(by Leonard Schiff) mentions the following function as the...

This is very simple question, and i just need a 2nd opinion. We have a Space Station (preferably a torus) with angular velocity ω and radius r. We have a car inside which OPPOSES the angular velocity and moves with the speed ωr . So, will the gravity felt in this car be Zero? Or will it be...

Most number theorists will be familiar with the result conjectured in the 19th century and proved in the 20th century that the only square pyramidal numbers that are square numbers are 1 and 4900 (the sum of the squares from 1^2 to 24^2 = 70^2). While discussing this, it was pointed out to me...

Hello, I am reading a paper on evolution problems of the type u(0) = 0 u' (t) = v (t) v(t) = argmin {G (t, u(t), v(t)} The authour claims that the Euler Lagrange equation related to the last equation is dG / dv = 0 (1) I can not understand this. The Euler Equation...

Homework Statement y'' + y' = 4t Homework Equations Use Variation of parameters! The Attempt at a Solution So I get homo of: c1 + c2 e^-(t) From there I get a Wronskian of -e^(-t) Then I get variations 2t^2 and -4e^t(t-1) Then get the answer of 2t^2 + 4t - 4 Btu...

Homework Statement Find the particular solution to t^2 y'' - t(t + 2)y' + (t+2)y = 2t^3 given that y1 = t and y2 = tet are solutions. Also, require that t > 0 The Attempt at a Solution Rewrite the original equation as y'' - ((t + 2)/t)y' + ((t+2)/t^2)y = 2t So first I calculate the...

Hi folks, In a followup to another thread about the recent controversial results by Webb et al. suggesting that the fine structure constant (alpha) varies across the visible universe, I was wondering: Assuming the Webb et al. results are correct (a big assumption), would we be able to say...

Homework Statement Find the particular solution to t^2 y'' - t(t + 2)y' + (t+2)y = 2t^3 given that y1 = t and y2 = tet are solutions. Also, require that t > 0 The Attempt at a Solution Rewrite the original equation as y'' - ((t + 2)/t)y' + ((t+2)/t^2)y = 2t So first I calculate the...

Hey, The question is displayed in the image below: So I have approached this question using g=-GM/(r^2), for the surface and the Earth as a whole. Though when it talks about the average density I wasn't sure if it meant the whole Earth or a sphere at a smaller radius (that of the...

Hey guys. In my mechanics course, we have began discussing calculus of variations, and I don't really understand what's going on, entirely. Any help understanding would be great. Our professor gave us an easy problem, but I feel like I am just missing something. Homework Statement...

NOTE : My Current question and source of debate is in post number 10. Lets say I have a square coil. I accelerate it in a direction such that it is perpendicular to a magnetic field directed into the page. As it enters the field and before it is completely in the field an...

This question about the variation of the aforementioned quantities in a transformer. Now according to me and my knowledge of electricity this is how I feel it should play out : If the input current were a sine curve that varies with time then : Input voltage = cosine curve i.e. it is...

Hi, I've come across a question in a stats book which asks the following: Q: A study was undertaken to find the relationship between "emotional stability" and performance in college. The following results were obtained: Emotional stability, Mean = 49, Standard Dev = 12 College Average, Mean...

Homework Statement Original Monty Hall problem: There are 3 doors, 1 of them contains a car and the other 2 goats. You choose 1 door, the host opens a door that is not chosen by you and does not contain the car. Then you can change to the other closed door, or keep your own chosen door at...

Homework Statement 2. About 1 in 200 Amish are homozygous recessive for Ellis-van Crevald syndrome, which causes short stature, extra fingers and toes, and several other characters. A) What % of Amish people are likely to be heterozygote carriers for the disorder? B) If the...

Hi, I'm not exactly sure how to solve the following non-homogeneous ODE by variation of parameters. Solve the given non-homogeneous ODE by the variation of parameters: x^2y" + xy' -1/4y = 3/x + 3x Can someone please point me in the right direction? Help will be much appreciated...

Hello new to this forum , Was solving some Diff eq problems and iam getting two different answers using two methods, ok the problem is i=primes (x^2)(y^ii)+(x)(y^i)+y=4sin(lnx) This is cauchy method, When i use variation of parameters i get a long answer with impossible integrals and when i...

Homework Statement Solve y''+25y=10sec(5t) Homework Equations NA The Attempt at a Solution I believe I have the correct answer for yp which is: 2/5log(cos(5t))cos(5t)+2tsin(5t) When I plug this into the Webwork field, it says it is incorrect. I checked my answer against...

Hey ya'll! This is the equation under discussion: y'' - 2y' - 3y = x + 2 I'm asked to use the method of variation of parameters to determine a solution for this differential equation, but I reach a point where my the equations just look too ridiculous to continue. The point I have in...