In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence
{
x
n
}
n
=
1
∞
{\displaystyle \{x_{n}\}_{n=1}^{\infty }}
of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.
Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces.
if a\inK be seperable element over the field F,let f(x) be minimal plynomial of a over F.let degree of f(x) be n, We know f(x) will be seperable over F, BUT is it necessary that f(x) will be MINIMAL POLYNOMIAL of other n-1 roots. if so, what is the reason?
Homework Statement
2y * y'(t) = 3t^2 such that y(0) = 9
Homework Equations
g(y)y'(t) = h(t)
The Attempt at a Solution
So I have done many of these seperable ones in homework that did not require a parameter so now I got lost.
This is what I did;[/B]
Integral of 2y dy = integral of 3t^2...
Homework Statement
dx/dt=x^2+1/25,
and find the particular solution satisfying the initial condition
x(0)=8.
Homework EquationsThe Attempt at a Solution
So I began by taking out 1/25 from the right side, making the equation:
dx/dt = (1/25)(25x^2 + 1)
Then, rearranging the equation to be...
Hello. I need some help solving a differential equation. I think where I'm going wrong is integrating one side via partial fractions, but I'm not quite sure where my mistake is. Using Wolfram, I found the correct answer, which is below. Thanks.
Homework Statement
Solve the following initial...
I have this equation: dy/dx = 1-y^2, so then dy/(1-y^2) = dx, so ∫dy/(1-y^2) = dx --->
∫(A/(1+y) + B/(1-y))dy = x + C. I rewrite it again: (A - Ay + B + By)/(1+y)(1-y) = 1/(1+y)(1-y) so I get A+B = 1, and B-A = 0, so B = A, and therefore 2A = 1. so A & B = 1/2.
So 1/2∫(dy/(1+y) +...
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...
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...
( 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...
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...