Variation Definition and 540 Threads

Homework Statement Solve using variation of parameters y''' - 2y'' - y' + 2y = exp(4t) Homework Equations Solve using variation of parameters The Attempt at a Solution I got the homogenous solutions to be 1, -1, and 2. So, y = Aexp(t) + Bexp(-t) + Cexp(2t) + g(t) I got...

The question I'm trying to solve is: y" - 6y' + 9y = \frac{exp(3x)}{(1+x)} I formulated the Gen solution which are: y1(x) = exp(3x) and y2(x) = xexp(3x) I've then calculated the wronskian to get: exp(6x) I then went onto to use the variation of parameters formula, which is where...

I just realized you can use variation of parameters (VOP) to solve for homogeneous 2nd order equations. I see it takes much longer to do so. But I was wondering why, if you use VOP, the u and v functions are 0. Is this because the coefficients of the homogeneous equation are constant, or...

Homework Statement "Vary the following actions and write down the Euler-Lagrange equations of motion." Homework Equations S =\int dt q The Attempt at a Solution Someone said there is a weird trick required to solve this but he couldn't remember. If you just vary normally you get \delta...

Find the complementary solution of y^\left(4\right) + 2y'' + y = sint Homogeneous Form would be y^\left(4\right) + 2y'' + y = 0 r^4 + 2r^2 + r = 0 \rightarrow r(r^3 + 2r + 1) = 0 This is where I'm stuck. Once I find y_c(t) I should be able to finish the problem, but I'm having trouble at this...

Hi everyone, i have a question abuot how my professor is using the pressure variation equation and I would really appreciate help with it! Homework Statement How high can you suck water up a strw? The pressure in the lungs can be reduced to about 10 kPa below atmospheric pressure 2...

Does youngs modulus of elasticity depend upon temperature?

Homework Statement I must solve (1-x)y''+xy'-y=(1-x)^2 knowing that y=x is a solution if the right hand side is 0. I must use this fact in order to obtain the general solution to the DE Homework Equations Variation of parameters? The Attempt at a Solution I'm looking at...

Homework Statement The most commonly used algorithm for computing \sqrt{a} is the recursion xn+1 = 1/2 (xn + a/xn), easily derived by means of Newton's method. Assume that we have available to us a very simple processor which only supports addition, subtraction, multiplication, and halving (a...

I read the following in Fowles & Cassiday's Mechanics: "The correct motion that a body takes through space is that which minimizes the time integral of the difference between the kinetic and potential energies" or \deltaJ = ∫ L dt = 0...

I've picked up a bit more since my last problem. I need to solve the following DE: x^{2}\frac{dy}{dx}+x(x+2)y=e^{x} I decided to use variation of parameters, so I re-arranged it like so: \frac{dy}{dx}=\frac{e^{x}}{x^{2}}-(1+\frac{2}{x})y Then solved the homogenous DE...

Can somebody clarify how the formula for variation of the auxilliary worldsheet metric is obtained due to reparametrization of the worldsheet in string theory??

I need to find a solution to the following problem: (x^{2}-1)\frac{dy}{dx}+2y=(x+1)^{2} y(0)=0 I decided to try using variation of parameters. My teacher was unable to show any examples, and I'm having issues understanding the textbook. From what I see I need to get it onto this form...

Homework Statement Pascal's law states that " The pressure in a fluid at rest is the same at all points if they are at the same height" Also we know " Pressure increases with depth" I get confused. When pressure increases with distance, how pressure is same at all points. Homework...

[b]1. Homework Statement [/ The value of acceleration due to gravity (g) at an altitude (h) is gh = g (1 - 2h/R). Similarly the value of g at a depth (d) is gd = g(1 - d/R), where R is the radius of the earth. Homework Equations In both the cases, my book says the value of g decreases...

Homework Statement Using the variation of parameters method, find the general solution of x^{2}y" - 4xy' + 6y= x^{4}sin(x) Homework Equations y_{P}=v_{1}(x)y_{1}(x) + v_{2}(x)y_{2}(x) v_{1}(x)'y_{1}(x) + v_{2}'(x)y_{2}(x)=0 v_{1}(x)'y_{1}(x)' + v_{2}'(x)y_{2}(x)'=x^{4}sin(x)...

Given an action: S = \int L(q,\dot{q},t) \,dt The variation is: \delta S = \int \left(\frac{\partial L}{\partial q}\delta q+\frac{\partial L}{\partial \dot{q}}\delta\dot{q}\right)\,dt I'm guessing this is some type of chain rule, but I haven't been able to derive it... how is it...

Homework Statement Find a particular solution for these second order differential equations. Homework Equations 1) y''+9y=tan3t 2) y''+y=tan^2t The Attempt at a Solution I want to find a fundamental solutions y1 and y2 because I want to find a particular solution like this...

Homework Statement Solve for general solution with variation of parameter y'''(x) - y'(x) = x The Attempt at a Solution I initially looked at y'''(x) - y'(x) = x only and I foudn my answer to be y(x) = C_1e^{x} + C_2e^{-x} + 1 - x Now i looked through my book and it says it works for...

Homework Statement The Schrodinger equation (in atomic units) of an electron moving in one dimension under the influence of the potential -delta(x) [dirac delta function] is: (-1/2.d2/dx2-delta(x)).psi=E.psi use the variation method with the trial function psi'=Ne-a.x2 to show that...

If a 1kg object is moving at 3m/s in a positive direction, and a 12N force is applied in the negative direction, what is the velocity immediately after 2s? I'm fairly sure this will be a variation of relevant momentum equations, and/or mixed with kinematics, yet I'm not seeing the correct...

My first post, yay i already like the atmosphere here :P anyway... Using the formula F = kQq/R2 sketch graphs between a. F and Q (k,q, and R are constant) b. F and R (Q,q and k are constant) c. Q and R (k,q and F are constant) I think i did it correctly but I'm not quite sure...

I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a...

Hi. During daytime sea water having poor solar reflectivity remains warmer than the air. But at times, water has also been found to be colder than air, with the difference being 5-10 degrees C. Can anyone please justify how could that be possible?

Work done by a gas = PV But when we derive specific heat of the gas at constant volume, even though the pressure changes we take work W=0. Or in the case where both pressure and volume are changing and we want to find work done we take the integral of d(PV) where we replace P with nkT/V. Also...

Am I right in thinking that the statement "f:[a,b]-->R is of bounded variation" is equivalent to the statements "f:[a,b]-->R has bounded range" and "f;[a,b]-->R is a bounded function".

Given t^2 y'' -t(t+2)y' = (t+2)y= 2t^3 and y1= t, y2= te^t Find the particular solution- I ve worked the problem to [ -2t^2 -2t] by: -t * Integral [ 2t* te^t/ t^2e^t] + te^t * Integral [ 2t^2/ t^2e^t] whereas the book states that it is simply -2t^2. Can you guys tell me where I made...

So I'm doing some practice problems to prepare for a test on Friday and I'm just curious about this problem:: y'' + 3y' + 2y = 4e^(x) in factoring using characteristics: (r+2)(r+1) = 0 r = -2,-1 so Yc = C1*e^(-2x) + C2*e^(x) y1= e^(-2x) y2= e^(-x) (skipping some algebra)..I...

Hi All, is there anybody to give me some help on how I can calculate the Euler Lagrangian equation associated with variation of a given functional? I am new with these concepts and have no clue about the procedure. thanks a lot

Hello! I have a question? Does anyone know if scientists have tried to run the electrons through the double slit, record the particles that go through each slit, and before looking at the results, looking at the pattern on the screen. And THEN doing the EXACT OPPOSITE of what the screen...

I can't get the variation of formula http://img813.imageshack.us/img813/3754/38919739.png in the form of [PLAIN][PLAIN]http://img839.imageshack.us/img839/536/96608635.png. Can anyone help me. Sorry, I am not good at math :)

Hello, in the paper from sean carroll "the cosmological constant" we can read this: Does this variation of the cosmological constant after symetry breaking is considered as real and accepted in standard cosmology? I find very few talks about a varying cosmological constant, and it is about...

Homework Statement \mathbb{x'}= \begin{bmatrix} 1 & 1 \\ 4 & 1 \end{bmatrix} \mathbf{x} \ + \ \begin{bmatrix} 2e^{t} \\ -e^{t} \end{bmatrix} Find the general solution. Homework Equations The Attempt at a Solution Well i found the eigenvalues of the matrix That i'll call...

Homework Statement A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (1) Rabbit pairs are not fertile during their first two months of life, but thereafter give birth to three new male/female pairs at the end of every month...

Hi everyone! Here's my problem: Let's suppose that we have a functional I[f,g]=\int{L(f,\dot{f},g,\dot{g},x)\,dx}. Is it right to say that the variation of I whit respect to g (thus taking g\;\rightarrow\;g+\delta g) is \delta I=\int{[L(f,\dot{f},g+\delta g,\dot{g}+\delta \dot...

Homework Statement An enclosed vertical tube rotates about its vertical axis at w=3000rpm. At the axis, r=0, P=1.5bar and T=293K. What is the pressure distribution as a function of r? And hence calculate the pressure at r=2m. The Attempt at a Solution I have seen this type of questions...

ty''-(t+1)y'+y=t^2 I know I have to use variation of parameters to solve this. But I am stuck and cannot figure out how to get the homologous equation! y''-(1+\frac{1}{t})y'+\frac{1}{t}*y=t I don't know how to solve this homologous equation in this format. Is it R^2+(1+1/t)R+1/t = 0 ? How...

Homework Statement If a function f is of bounded variation on [a,b], show it is Riemann integrable Homework Equations Have proven f to be bounded S(P) is the suprenum of the set of Riemann integrals of a partition (Let's say J) s(P) is the infinum of J S(P) - s(P) < e implies f...

I am working on a problem requiring variation of parameters. When I calculated the wronskian, I got an answer, which differed from the book only by a "-" (mine was -, the book's was +). So I switched my functions for y1 and y2 and got the answer the book had. Is there a standard for which...

I have already read one thread on Lagrange's method of variation of parameters and it was very useful, but I am still confused about the use of the constraint. If the solution to the homogeneous second order equation contains two functions, with arbitrary constants: y= Ay1 + By2...

Homework Statement A gambler has 2$ and wants to have 10$. To get the money he enters a game where a fair coin is tossed. If he bets on the right side he wins doubles his stake and if he bets wrong he loses his stake. The strategy is to bet everythig if he has 5$ or less and just enough to walk...

I am trying to solve a problem along the lines of y'' + 2y' + y = e^(-x) (2 + 1/x^2).. The actual one I am trying to solve differs slightly. I was trying to solve it using the method of variation of parameters.. However it is new to me and was too confusing. So first I get: y comlpiment...

Homework Statement Find a particular solution by method of variation of parameters: t2y'' - 2y = 3t2 - 1 given: y1 = t2 y2 = t-1 Homework Equations The Attempt at a Solution I get Y(t) = t^2ln(t) - \frac{1}{3}t^2 + \frac{1}{2} The book gives Y(t) = t^2ln(t) +...

I was looking through my DE book and a problem intrigued me. I eventually figured it out but I do not understand the logic. I was wondering if anyone here could help me out. The question says: Use the method of variation of parameters to show that...

in varying an action like Polyakov's action with respect to the metric on the world sheet we have to consider the variation of the square root of the determinant. I have not found how to express the variation of the determinant of the metric. From reverse engineering I found that \delta(h)=2...

I have a question about calculus of variation. does anybody here know a proof for the chain rule: \delta S= \frac{dS}{dx} \delta x and for the formulation: \delta S= p \delta x => \frac{dS}{dx}= p it would be totally sufficient, if anyone here knows(e.g. a weblink) where one could see this...

Homework Statement Alright, so I just want some clarification on direct variation, since it seems that every internet source I can find is (seemingly to me) wrong. To me, direct variation means that the ratio of y to x is fixed with y=kx where k is the constant of proportionality...

hey I do not understand a step here! The integral is: \delta S(x,t)=-mc \int_a^b u_i d \delta x^i =0 and now they say one should do integration by parts, but I do not know how this should work here? Where are my two functions?As far as I see there is only the four-velocity and I do not how...

1. Homework Statement [/b] If f has a continuous derivative on [a,b], and if P is any partition of [a,b], show that V(f,P)\leq \intablf'(t)l dt. Hence, Vba\leq\intablf'(t)ldt. Homework Equations Monotone function \subset BV[a,b] \sumf(ti+1)-f(ti) = lf(b) - f(a)l The Attempt at a...

Homework Statement Given a sequence of scalars (cn) and a sequence of distinct points (xn) in (a, b), define f(x) = cn if x = xn for some n, and f(x) = 0 otherwise. Under what condition(s) is f of bounded variation on [a,b]? Homework Equations Vbaf = supp(\Sigmalf(ti) - f(ti-1)l< +inf...