Integral equation Definition and 102 Threads

The textbook, "Mathematics for Physical Chemistry - Opening doors" (McQuarrie), solves this example excercise as follows: And is the case for n =/= m which I'm troubled with, because, if I solve the integral instead of using the cycles argument, I get that: And I can't see how this is equal...

I have some variables that are uncertain, these are w_m = u.ufloat(0.1430, 0.0011) z_rec = u.ufloat(1089.92, 0.25) theta_srec = u.ufloat(0.0104110, 0.0000031) r_srec = u.ufloat(144.43, 0.26) and some constant values c = 299792.458 # speed of light in [km/s] N_eff = 3.046 w_r = 2.469 *...

Hello. I need help in simplifying this integral equation, i know the final result must be 2(1-x)^1/2 + C. I been stuck on this one for a while.

Thanks

I did the first part, it is part (b) that I'm having trouble understanding. For any ##x \lt b##, ##f(x)=0## and ##\int_0^x {f(t)} \, dt = 0## (since ##f## is 0 everywhere from 0 to ##b##), which turns the equation ##\int_0^x f(t) \, dt = (f(x))^2+C## into ##0=0+C##, which implies ##C=0##. But...

I am able to solve the problem however if x was position and t was time how is this problem interpreted? I know, for example that ##\frac{dx}{dt}## tells us how the position of something changes as time changes (or instantaneous change) and an integral gives a net change so to speak.

We often neglect the terms of a surface integral ##\int_v(\nabla•A)dv=\int_s(A•ds)## for ##s## to be very large or ##v## to be very large, What is actually the reason behind this to neglect??

The book on quantum mechanics that I was reading says: d<x>/dt = d/dt ∫∞-∞ |ψ(x,t)|2 dx =iħ/2m ∫∞-∞ x∂/∂x [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (1) =-∫∞-∞ [ψ∂ψ*/∂x+ψ*∂ψ/∂x]dx (2) I want to know how to get from (1) to (2) The book says you use integration by part: ∫abfdg/dx dx = [fg]ab - ∫abdf/df dg dx I chose f...

Homework Statement The question is; for a qunatum mechanical particle, Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h] in here, x0, p0 and h are constants, so, Homework Equations what are the <x>; expetation value, and <P>;expectation value of momentum ? The Attempt at a Solution , [/B]...

The question is; for a qunatum mechanical particle, Ψ(x) = [1/(a1/2.π1/4)].[e-(x-xo)2/2a].[eip0x/h] in here, x0, p0 and h are constants, so, what are the <x> and <P> ?

1. At pg.212, Hartle book (2003) writes equation 9.81 as an approximation of 9.80, directly. 2. $$ΔΦ=\int_0^{w_1}\frac{(1+\frac{M}{b}w)}{(1+\frac{2M}{b}w-w^2)^\frac{1}{2}}dw$$ equation(9.80) $$ΔΦ≈\pi+4M/b$$...

Hi! I would like solve this kind of relation: \phi = \int_0^r \phi (r') 4 \pi r'dr' But I don't know how to proceed... Can you advise me ? Thank's in advance !

Hello, I just want to clarify some things with a simple exercise: I have the equation ## \frac{\partial^2 f}{\partial A^\mu \,\partial A^\nu} = 0## and I want to integrate it once assuming that ## f=f(A^1,A^2,...,A^n)=f(A^\rho) ##. I think the solution should be ## \frac{\partial f}{\partial...

Homework Statement A specific problem of the Fredholm integral equation is given as $$\phi(x) = \pi x^2+\int_0^\pi3(0.5\sin(3x)-tx^2)\phi(t)\,dt$$ and the exact solution is ##\phi(x) = \sin 3x##. Homework Equations Nothing comes to mind. The Attempt at a Solution I'm unsure how to approach...

Homework Statement find f(x) which satisfies f(x) = x + ##\frac{1}{\pi}## ##\int_{0}^{\pi} f(t) \sin^2{t} \ d(t)## Homework EquationsThe Attempt at a Solution to solve f(x), I have to solve the integral which contains f(t). And f(t) is the f(x) with variable t? if yes, I will get integral...

Homework Statement The problem states: "A point charge q is located at a fixed point P on the internal angle bisector of a 120 degree dihedral angle between two grounded conducting planes. Find the electric potential along the bisector." Homework Equations ΔV = 0 with Dirichlet boundary...

Suppose I have an infinitely thin foil that absorbs 1% of incoming radiation. It therefore emits 0.5% of the incoming radiation in both directions perpendicular to the plane of the foil. That is, transmission is 0.5%. How to calculate transmission for a series of such foils? There will be a lot...

I have the following expression : $$ y_{E} = \int_{0}^{\infty} 0.5 * [E_{1}(µ(E)*r) - E_{1}(\frac{µ(E)*r}{cos \alpha})] * f(r) dr $$ where : - $y_{E}$ has been measured for some E (something like 5 different $E_{i}$, to give you an idea) - µ(E) is retrieved from a table in the litterature...

I'm following a derivation in my lecture notes of total average particle number in an ideal classical gas (statistical physics approach). I follow it to the line (though the specific terms don't matter): \left<N\right> = e^{\mu/\tau} \frac{\pi}{2} \int_0^\infty \left(n \,dn \,e^{- \frac{\hbar^2...

I would like to solve an equation below using MATLAB: All the parameters except p are known, so I only need to solve for p. However since I need to consider the sign of the integrand and there is an absolute value sign in it I don't know how to solve it. Could anyone please help? Thank you.

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...

Homework Statement Find a continuous funciton ##f## such that $$ f(x) = 1+ \dfrac{1}{x} \int_{1}^{x} f(t)dt $$ I think I solved it but I would like to see if it's right. Well, first of all, by the fundamental theorem of calculus I know that $$ \left( \int_{1}^{x} f(t)dt \right) ' = f(x) $$...

Hello! Could you tell me about how to take the next numerical calculation in mathematica? (perhaps there are special packages). I have an expression (in reality slightly more complex): ## V=x^2 + \int_a^b x \sqrt{x^2-m^2} \left(\text Log \left(e^{-\left(\beta...

The problem I want to find ##x## which solves ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0 ## The attempt ##\int^x_1 \frac{\sin t}{t} \ dt = x -1 ## I see that the answer is ##x=1## but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any...

Please anyone can help solve this integral equation e^t+e^t ∫ (t, 0 ) e^(-τ) x f(τ) dτ

I had posted a question earlier which this is related to, but a different equation. $$\frac{d}{dt} \int_0^t H(t,s)ds = H(t,t) + \int_0^t \frac{\partial H}{\partial t}(t,s)ds$$ This was another formula needed in a proof however I don't see how this one holds either. I tried following a proof of...

Hello! How do I solve the following integral and differential equations 1. $x^2f''(x)-2xf'(x) = x$ 2. $\int_0^{x} (1+f(x))\;{dx} = x$ I'm supposed to http://mathhelpboards.com/linear-abstract-algebra-14/linear-subspaces-17957.html all the polynomials that satisfy this, but I can't. (Shake)

Homework Statement Particular solution of y" - y' - 2y = e^(2x) Homework Equations None The Attempt at a Solution This makes no sense to me, why do I have to use the solution of the form y(t) = cxe^(2x) For the problem above, but when I switch the signs and it becomes y" - y' + 2y =...

Homework Statement Hi - this exercise appears in a section on Fourier Transforms. Show ## \int e^{ik \cdot (r - r')} \frac{d^3 k}{(2 \pi)^3 \vec{k}^2} = \frac{1}{4 \pi}|r-r'| ## Hint: use spherical coords in k-space. Homework Equations I am not sure of the coord. transform, but from the...

Recently I was a witness and a minor contributor to this thread, which more or less derailed, in spite of the efforts by @Samy_A. This is a pity and it angered me a bit, because the topic touches upon some interesting questions in elementary functional analysis. Here I would like to briefly...

Hello everyone! I am building set of Fortran code to solve integral equation. I have read "Numerical recipe" and heard about "Nystrom method". But there's no sample problem, I found it difficult to understand. Can anyone explain "Nystrom method" for me with a simple problem? Many thanks

Have added attachment. Can anyone show me how to approach this problem? Thank you...

How to prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$ To prove that $\int_0^\pi {x\,f(\sin x)\,} dx = \frac{\pi }{2}\int_0^\pi {f(\sin x)} \,dx$ is true, first I started calculating the integral of the left indefinitely $$ \int {x\,f(\sin x)\,\,dx} $$...

if , then what will be . In fact I was solving the integral equation by the method of successive approximation.

Homework Statement Ok guys, its been a while since I have done this one so I don't even know where to begin. Here is the problem I have to complete: The power (in watts) from an engine is represented by the equation below, where t is the time in seconds. P = 30t^2.2 + 3t 1) Draw a table...

Homework Statement The volume of the solid obtained by rotating the region bounded by x=6y^2 http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmmi10/alpha/144/char3B.png y=1 [PLAIN]http://msr02.math.mcgill.ca/webwork2_files/jsMath/fonts/cmmi10/alpha/144/char3B.png...

Need help showing that both sides of the following integral are equal. Any help would be greatly appreciated.

Is it possible to convert a general linear second order boundary value ode y'' + P(x)y' + Q(x)y = g(x), y(a) = y_a, y(b) = y_b to a Fredholm integral equation, explicitly determining the Kernel in the process, without removing the y' term? (Here is an example of doing it without the y'...

Homework Statement Let the curve C be given by ##\vec{r}(t)=3t^{2}\hat{\imath}-\sqrt{t}\hat{\jmath}## between ##0 \leq t \leq 4##. Calculate ##\int_{C} xy^{2}dx+(x+y)dy##. Homework Equations The Attempt at a Solution First find the derivative of r...

I have to solve the integral equation $$y(x)= -1+\int_0^x(y(t)-sin(t))dt$$ by the method of successive approximation taking $$y_0(x)=-1$$. Sol: After simplification the given equation we have $$y(x)=-2+cos(x)+\int_0^x y(t)dt $$. So comparing it with $$y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt$$...

I have to solve the integral equation $$y(x)=1+2\int_0^x(t+y(t))dt$$ by the method of successive approximation taking $$y_0(x)=1$$. Sol: After simplification the given equation we have $$y(x)=1+x^2+2\int_0^xy(t)dt$$. So comparing it with $$y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt$$ we have...

Homework Statement Hi. As you all know, ##<F> = <F>^*##, where ##F## is an observable linked with the operator ##\hat{F}##. This means ## <F> = \int \Psi^* \hat{F} \Psi d\tau = <F>^* = [\int \Psi^* (\hat{F} \Psi) d \tau]^* = \int \Psi (\hat{F} \Psi)^* d \tau \Rightarrow \int \Psi (\hat{F}...

Homework Statement Hi everyone. I have encountered a weird equation while doing some research and I have no idea how to solve it. The equation goes like this ∫ dR / (1+ c*r) ^ (a/r) = d, limits of integration are from 0 to Rmax, where Rmax ^2 = [u]^2 - α^2, where u is a constant value...

Given \[ f(x) = 1 + \lambda\int_0^1(xy + x^3y^2)f(y)dy. \] To use the Fredholm series, we need to find \(D(\lambda)\) which is \(1 - \text{order }\lambda + \cdots\). I have already calculated the orders and of order 3 and 4 the terms are 0 so the terms greater than 3 are all zero since we have...

Given \[ f(x) = \lambda\int_0^1xy^2f(y)dy \] I am trying to determine the eigenvalues and eigenfunction. I know that the \(\frac{1}{\lambda}\) are the eigenvalues. We can write \(f(x) = xA\) and \(A = \lambda\int_0^1y^2f(y)dy\). \[ A\Bigg(1 - \lambda\int_0^1y^3dy\Bigg) = 0\quad (*) \] So is...

I need to solve an integral equation of the form $$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$ where - f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$ - K: [0,1] x R -> [0,1] is the unknown kernel I am looking for a solution other than...

Homework Statement If \displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n} = 6 and a+b=16, then find a and b. The Attempt at a Solution \displaystyle \int^b_a \dfrac{dx}{1 + (16/x - 1)^n} = 6 I tried substitution but it did not work.

Hello, I have been encountered an integral equation that I need to solve\evaluate numericly and I didn't find anything like it in my search yet. The equation: \frac{d^2 F(t)}{dt^2}=const*\int_{t'}\frac{sin(F(t)-F(t'))}{(t-t')^{2k}}dt' If it helps there is a specific case, when K=1...