Separable Definition and 185 Threads

The first postulate of quantum mechanics says that every physical system is associated with a separable complex Hilbert space, however this does not hold for a free particle, where the basis is uncountable (all the momentum kets). I think it also does not hold for a free falling particle...

Homework Statement \frac{du}{dt} = e^{5u + 7t} Solve the separable differential equation for u: Use the following initial condition: u(0) = 6. The Attempt at a Solution I tried to take the natural log of each side but now I'm stuck. How can I separate the equation when both the u...

Here is the question: I have posted a link there to this thread so the OP can view my work.

AP Physics student here, I'm working on a problem that takes into account air resistance, where something is thrown up at initial velocity v_0, and the drag force is proportional to the velocity, so, \vec{F_{drag}}=-k\vec{v}. Using Newtons second law and making up positive, down negative, you...

Homework Statement Find the solution to the differential equation. Which passes through the point (0,e). \frac{dy}{dx} = \frac{11xy}{(ln(y))^{10}} Homework Equations I can get through the integration part but I am lost when it comes to using the ln rules to find the solution...

Entanglement is inseparable state |ψ>=\Sigma\Sigmac(n1,n2)|n1n2> but for decoherence where system and environment coupling |ψ>(s,e)=\Sigmac|s>\otimes|e> which is normal tensor product, can we still say entanglement cause decoherece or just the tensor product between system and environment?

Homework Statement Solve the equation dy/dx = x/(y^2√(1+x)) Homework Equations The Attempt at a Solution I separated them: y^2 dy = dx/√(1+x) I then integrated the dy side, I got (1/3)y^3 + C. I am stuck at integrating the dx side. Thanks in advance!

Suppose I have a variable separable ODE, e.g., \frac{dy}{dx} = 3y. We all know that the solution is y=Ae^{3x} where A is a constant. My question is as follows. To actually find this solution we rearrange the equation and integrate to get \int \frac{dy}{y} = 3 \int dx, which gives \ln...

So I have a question in terms of interpreting the boundary conditions for a PDE. It is question 4 in the attached picture. My question is that usually when I have encountered BC problems I have been given that my boundary conditions equal a given value, in terms of the diffusion equation...

I haven't done ODEs in a while nor have a book handing. How do I tackle an equation of the form \[ 2xyy'=-x^2-y^2 \] I tried polar but that didn't seem to work.

\sqrt{1-y^2}dx - \sqrt{1-x^2}dy=0, y(0)=\frac{\sqrt{3}}{2} rewriting the equation gives \frac{1}{\sqrt{1-x^2}}dx = \frac{1}{\sqrt{1-y^2}}dy Isn't this the integral for \sin^{-1}(x) & \sin^{-1}(y)? The back of book has y=1/2x+\frac{\sqrt{3}}{2}\sqrt{1-x^2}

\frac{dy}{dx}+2xy=0 \frac{dy}{dx}=-2xy dy=-2xy dx \frac{1}{y} dy=-2x dx integrate both sides \ln{|y|}=-2x+c y=e^{-2x+c}=e^{-2x}e^C=e^{-2x}k=ke^{-2x} Let's check using the original equation. First calculate the derivative \frac{dy}{dx}=k(-2e^{-2x}=-2ke^{-2x} so from the original...

Homework Statement Solve d2y/dt2 = dx/dt2, if x = 0 and dx/dt = 1 when t = 0 Homework Equations The Attempt at a Solution d2y = dx I'm not exactly sure what to do here the fact that dt2 is under the denominator for both fractions is confusing memaybe its a typo? should it be d2y/dx2 = dx/dt?

Homework Statement Show that if X\subsetM and (M,d) is separable, then (X,d) is separable. [This may be a little bit trickier than it looks - E may be a countable dense subset of M with X \cap E = Ø.] Homework Equations No equations, but there are relevant definitions. Per our book: A metric...

Homework Statement . Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable. The attempt at a solution. I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y: X is separable...

Homework Statement Solve the separable differential equation for u du/dt=e^(5u+2t) Use the following initial condition: u(0)=13 The Attempt at a Solution Honestly I didn't get very far on this one. I took the natural log of both sides, ln du/dt = 5u+2t And now I am stuck. Should I divide...

Homework Statement Here is a photo of a page in Laser Physics by Hooker: https://www.evernote.com/shard/s245/sh/2172a4e7-63c7-41a0-a0e7-b1d68ac739fc/7ba12c241f76a317a6dc3f2d6220027a/res/642710b5-9610-4b5b-aef4-c7958297e34d/Snapshot_1.jpg?resizeSmall&width=832 I have 3 questions (I'm a bit...

I have read that, if you given a differential equation \frac{dy}{dx} = f(x,y), and can write it in the form \frac{dy}{dx} = h(x)g(y), then you can proceed with the following steps: \frac{dy}{g(y)} = h(x)dx integrating G(y) = H(x) + c Why are these steps vaild? I thought that one was not...

Here is the question: I have posted a link there to this topic so the OP can see my work.

( 4*x+1 )^2 dy/dx = 27*y^3 I'm trying to separate this into a separable equation. Does it matter which way I do it? I.e taking all xs to the left or all ys to the left or does it not matter as long as x and y are on different sides?

Greetings- In trying to solve a thermal stress problem, I have encountered an inhomogeneous differential equation of the following general form: \nabla^2 \Phi(r,z) = F_r(r)F_z(z) Solving the homogeneous case is no problem, as it is kind of a classic. Is there a route to finding a particular...

I have a sense that the countable, dense subset I'm looking for is the step functions, maybe over intervals with rational endpoints, but I'm not sure how to deal with the fact that E is any L-msb set, so there's no guarantee all the intervals are in there.

Here is the question: Here is a link to the question: General solution of dy/dt=k((y)(b-y))? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

∫ y' dx = ∫ dy ∫ yy' dx = ∫ y dy I can't clearly visualize how this is working, and it's messing with me when I work with equations that flip between displacement, velocity, and acceleration. Could someone link me to a proof, or maybe explain it a little for me? Thanks!

Homework Statement I am asked to find a singular solution of the D.E. dy/dx = (xy+2y-x-2)/(xy-3y+x-3). I am first solving to find the general solution form of the D.E., and so far have it to: [(x+2)/(x-3)]dx = [(y+1)/(y-1)]dy From here, of course, you integrate both sides, but I am...

I'm having difficulty with the question in the pic provided. (http://i.imgur.com/Fg7CHoY.png). Basically the equation I am given is dy/dx = (y-4x)/(x-y) and it walks through the steps needed to solve it, however, I am supposed to show how to arrive at each step given. I've gotten part a...

Suppose that we have rigged Gilbert space Ω\subsetH\subsetΩ\times (H is infinite-dimensional and separable). Is the Ω a separable space? Is the Ω\times a separable space? Consider the complete set of commuting observables (CSCO) which contain both bounded and unbounded operators...

The problem is dx/dt = (x+9)^2. This is separable so I made it dx / (x+9)^2 = dt. The only method I can think of using for something like this is partial fraction, but I can't get it to work with A/(x+9) + B/(x+9). Can anyone find a method that works?

Homework Statement dy/dx = (y^2 - 1)/ (x^2 - 1) with initial condition y(2) = 2 Why is y = 1 and/or y= -1 not solutions? Homework Equations The Attempt at a Solution I am actually able to solve this differential equation but when I separate the equation according to x and y...

I'm just starting my DE class, although I've been familiar with separable DEs for a while. Although they're (so far) pretty straight-forward to solve, I don't really understand the theory behind seperable DEs. In calc 1, it was stressed that dy/dx is NOT a fraction that can be "taken apart."...

Suppose X is a normed space and X*, the space of all continuous linear functionals on X, is separable. My professor claims in our lecture notes that we KNOW that X* contains functionals of arbitrarily large norm. Can someone explain how we know this, please?

Hi, The final step of solving a separable ODE is to find a function, f, defined implicitly by a relation G(y) = H(x). Say G(y) isn't defined at y = a and H(x) isn't defined at x = b, it appears to me that when rearranging such a relation to put y in terms of x, the point at which G(y) isn't...

Homework Statement x\frac{dy}{dx} = 4yHomework Equations I'm not sure if there is a specific equation for these type of problems. My professor just says to separate the two different variables and then integrate them with respect to x.The Attempt at a Solution \frac{1}{4y} \frac{dy}{dx} =...

Homework Statement Solve the given differential equation by separation of variables. Homework Equations dP/dt = P - P2 The Attempt at a Solution This is no problem to "solve" except that Webassign (:cry:) wants to know the whole thing in terms of P. You end up with...

Homework Statement Let S(t) represent the amount of a chemical reactant present at time t, where t>= 0. Assume that S(t) can be determined by solving the initial value problem http://webwork.math.ncsu.edu/webwork2_files/tmp/equations/21/885ac2eff6f65b363662233870e25e1.png where a, K, and...

It's my understanding that the definition of the indefinite integral is: ∫f(x)dx = F(x) + C, where d/dx [F(x) + C] = f(x) and C is an arbitrary constant And while dx has meaning apart from the indefinite integral sign the indefinite integral sign has no meaning apart from dx. Adding an...

Homework Statement A model for the shape of a tsunami is given by \frac{dW}{dx} = W\sqrt{4-2W} where W(x) > 0 is the height of the wave expressed as a function of its position relative to a point off-shore. Find the equilibrium solutions, and find the general form of the equation. Use...

I'm reading the following article by Maxwell Rosenlicht: http://www.jstor.org/stable/2318066 (The question should be clear without the article, but I present it here for reference.) In the beginning of the article he discusses differential fields (i.e. a field F with a map F\to F...

Homework Statement step 1. 2((2/3)y^(3/2) = 2x^(1/2) + C1 step 2. (2/3)y^(3/2) - x^(1/2) = C, where C = 1/2C1 The Attempt at a Solution I don't understand the where C = 1/2C1 - what is that? I understand everything else, except that.

There is something that has been bothering me recently: that is, the distinction between a separable state and being part of an entangled state. To make my query concrete, consider: \left|\psi\right\rangle = \alpha \left|0\right\rangle + \beta \left|1\right\rangle and...

I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!" I would like to...

Separable equations: How do you know which variable to solve for? + extra question Homework Statement I attached a sample problem with variables u and t. How do I know what the answer should be at the end? In terms of u or in terms of t or it doesn't matter? Homework Statement I also...

Greetings everyone, I haven’t done any quantum in a while, and was reviewing my textbook, Griffiths Ed. 1. The form of the Schrodinger equation I’m using is: i\hbar\partial\Psi/\partialt = -\hbar2/2m * \partial2\Psi/\partialx2 + V\Psi The book says if V is a function of x only, then the...

Homework Statement \frac{dy}{dx} = y \sqrt{x} , f(9) = 5 The Attempt at a Solution \int dy/y = \int \sqrt{x} dx ln |y| = \frac{2}{3} x^\frac{3}{2} + c y = e^{\frac{2}{3}x^\frac{3}{2}} + C y = Ce^{\frac{2}{3}x^\frac{3}{2}} 5 = Ce^{\frac{2}{3}9^\frac{3}{2}} 5 =...

Please see below link for the two different styles of solving a separable equation. http://en.wikipedia.org/wiki/Separation_of_variables#Ordinary_differential_equations_.28ODE.29 Which one is more proper? Why? My DE teacher told me that strictly speaking it's wrong to use the first method...

I have a question that is stumping me. I'd be grateful on some assistance. Show that the substitutions $z= ax + by + c$ changes $y' = f(ax + by + c)$ into an equation with separable variables. Hence, solve the equation $y' = (x+y)^2$. (hint: $\int \frac{1}{(1 + u^2)}du = tan^{-1} u+c$) I...

y' = (x)/(1+2y) y(-1) = 0 trying to find the answer I do the following: multiply both sides by (1+2y) (1+2y) * dy/dx = x i subtract 1 from both sides.. but for some reason this is wrong? why? 2y * dy/dx = x - 1 2y dy = x-1 dx integrate.. y^2 = (x^2-x+C)/2 y = sqrt((x^2-x+C)/2) ) this...

I am having a problem. I think i went well in decomposing the partial fraction and integrating, however my answer leaves me concerned. please help if i have gone wrong. Solve: dy/dx + y^2 = y. after taking partial fractions, i simplified this to: (1/y + 1/ (1-y) ) dy = dx and i integrated...

Homework Statement Can someone please verify if I am solving this equation right. Homework Equations Please refer to attachment. The Attempt at a Solution Please refer to attachment.

Homework Statement Find the general solution of the differential equation y'=4t-ty^2 Homework Equations y'=4t-ty^2 The Attempt at a Solution I 'think' this question is pretty straight forward but I'm still not sure if I did it right or not. I have two question. One till the last step...