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.
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...
Homework Statement
This is a physics problem,
for which if the following three dimensional potentials would Schrodinger equation be separable
V=x2y + sin(z)
V= x2 +y +tan-1 (z1/2)
Homework Equations
(-h2/2m)(d2ψ/dx2 + d2ψ/dy2 + d2ψ/dz2 ) +v(x,y,z)ψ=Eψ
The Attempt at a Solution...
Homework Statement
solve y(xy+1)dx + x(1+x^2y^2)dy=0
The Attempt at a Solution
well, I substituted u=xy. Here is what I've done so far.
du = xdy + ydx -> xdy= du - ydx -> xdy = du - (u/x)dx
(u/x)(u+1)dx + x(1+u^2)dy=0
(u/x)(u+1)dx + (1 + u^2)(du - (u/x)dx)=0
(u^2/x)dx + (u/x)dx +...
Hi guys,
I was working on this problem regarding separable equations and could not solve it..
dy/dx = ay+b/cy+d
My work:
I reorganized the equation to become dy(cy+d/ay+b) = dx
integrating both sides, you get the integral of (cy+d/ay+b) and dx which is a constant k.
I'm pretty sure...
Homework Statement
If y(1+x2) dy/dx = 2x (1-y2), prove that (1+x2)2(1-y2)=A, where A is constant.
Homework Equations
Separable equations
The Attempt at a Solution
Separate the terms:
y/(1-y2) dy = 2x/(1+x2) dx
Integrating both sides will get:
∫ y/(1-y2) dy = ∫...
Hello experts!
I need your point of view about the following,
Do you think is there any loss of generality if the arbitrary constant added when a separable equation is integrated is written in the form lnC rather than just C?
Do you think this would ever be convenient thing to do?
Is...
Dear readers,
Let X be the product space of a countable family \{X_n:n\in\mathbb{N}\} of separable metric spaces.
If X is endowed with the product topology, we know that it is again separable. Are there other topologies for X such that is separable? Is there a natural metric on X such that X...
2r(s^2+1)dr + (r^4 + 1)ds = 0
2.book answer different than mine...book's answer: r^2 + s = c(1 -r^2 s)
3. 2r(s^2+1)dr =- (r^4 + 1)ds
-2r/r^4+1 dr = 1/s^2+1 ds
int -2r/r^4+1 dr = int 1/s^2+1 ds
with u substitution on left we have u = -2r, etc.
tan^-1 r^2 = - tan^ -1 s + c
tan^-1 r^2 +...
Homework Statement
Suppse dn/dt = 1/10n and n(1)=-2. Separate the differential equation, then integrate both sides.
The Attempt at a Solution
How do I do this? There is no 't' to 'separate' from the equation. Would it just be:
int[1/10n] = int[0t] ?
Homework Statement
Find the solution of the given initial value problem in explicit form and determine the interval in which the solution is defined.
\[x dx+ye^{-x}dy = 0\] with initial condition y(0) = 1
Homework Equations
The Attempt at a Solution
I solved the first part...
I have solutions for 2 problems but they are different from the ones my book provides. This may be due to some simplification they chose to do, but I am uncertain.
1) dy/dx = x2/y
ydy = x2dx
Integrate both sides and you get
y2/2 = x3/3 + C
My book gave 3y2 - 2x3 = C
2) dy/dx =...
QUESTION:
Solve the separable differential equation
dy/dx = sqrt(4y+64), Initial Condition: y(4)=9,
and find the particular solution satisfying the initial condition.
MY ATTEMPT:
(dy/dx)^2 = 4y+64
((dy/dx)^2)-4y = 64
,/' (((dy/dx)^2)-4y) dx = ,/' 64 dx
Is this the right method...
Hi I have to solve the following LSQ problem:
min(||Aax-b||2)
where
A is a known matrix,
b is a know vector
x is an unknows vector
a is an unknown scalar
I can solve directly via pseudo inverse
ax=inv(A'A)A'b
but how can I isolate a from x?
Could Separable least square a scheme...
Homework Statement
dy/dx = e^ysin^2x/ysecx
Stewart 6e 10.3 # 8
Homework Equations
The Attempt at a Solution
ydy/e^y = sin^2xdx/secx
e^-ydy = sec^-1xsin^2xdx
Integration by parts
u = e^-y
du = -e^-y
dv = ydy
v = y^2/2
∫udv = e^-yy^2/2 + ∫y^2/2e^-y
= y^2/2e^y +...
Homework Statement
dy/dz = ycosx/(1+y^2), y(0) = 1
Stewart 6e, 10.3 # 12
Homework Equations
The Attempt at a Solution
∫(1+y^2)dy/y = ∫cosxdy
-------------- = sinx + C
How do I find the integral of this product? Do I use integration by parts?
Homework Statement
dz/dt + e^(t+z) = 0
Homework Equations
The Attempt at a Solution
dz/dt = -e^te^z
integral(dz/e^z) = integral(-e^tdt)
let u = 1/e^z
dv = dz
du = -e^-zdz v= z
integral(udv)
= z/e^z + integral(ze^-zdz)
Homework Statement
dy/dx = (y cos x) / (1+y2)
Homework Equations
Meh
The Attempt at a Solution
I've made it to this point:
ln(y) + y2/2 = sinx + C.
But we can't figure out to solve for y. It seems impossible with the ln(y) hanging around.
Hey.. I ran across some problems and the notation used is a little different from what I've seen before.
considering U(x,y)=X(x)Y(y)
Sometimes I'll see Uxx for \frac{d^{2}u}{dt^{2}} which equals X''Y
Or Ux for \frac{du}{dt} which equals X'Y
But what about U'x
Is that a redundant way of...
Homework Statement
I'm trying to solve this equation:
dN/dt = k(10 000 - N(t))
The attempt at a solution
dN/10 000 - N(t) = k dt
I integrate, and I'm left with:
- ln l10 000 - N(t)l = kt +C
I raise both sides with e. I don't know what e^- ln(xxx) is so I multiply both...
Homework Statement
Obtain the analytic expression for the N-channel T-matrix assuming a separable potential.
Hint: assume that T is proportional to V. Specialise your answer to N=1 and perform the required integral to get an explicit form for T, assume the given form for g(k)...
Problem and Equation: Solve dy/dx=-y/(x^2+y)
Put into standard form, this is ydx+(x^2+y)dy=0
The only ways of solving differential equations that I currently know are when they are either linear (which this is not), separable (this is also not), or exact (ditto), and I vaguely know about...
How do you find a solution for:
2tv' - v = 0
The text says it's separable but I'm not seeing it. I'm just learning so extra details are appreciated. Thanks.
(this should have been posted in the homework section - but I can't seem to move it there, sorry)
Hi all, I have a first order separable differential equation that I find a little difficult to solve. The question is from an old maths exam paper from my country. The book I obtained it from only has the answer, but I'd really like to know how to obtain the correct general solution. Okay, here...
Haven't done one of these in awhile and I was looking for a place to make sure I was doing it right. Hopefully one of you can take the time to look it over?
Homework Statement
Find the unique solution of the differential equation (3y^2)x(dy/dx)-x+1=0 for which y(e)=1
Homework Equations...
Homework Statement
1. Prove that if a metric space (X,d) is separable, then
(X,d) is second countable.2. Prove that \ell^2 is separable.
Homework Equations
The Attempt at a Solution
1. \{ x_1,\ldots,x_k,\ldots \} is countable dense subset. Index the
basis with rational numbers, \{ B(x,r) | x...
EDIT: I figured out my error, so don't worry about reading through all of this unless you find it an interesting problem
Homework Statement
This is Baby Rudin's exercise 2.27:
http://img63.imageshack.us/img63/584/fool.png
Instead of proving for R^k, I did it for an arbitrary separable...
stuck on separable equation (algebra prob??)
Homework Statement
y' = 2x/(1+2y) y(2) = 0
(1+2y)dy = 2xdx
integrate both sides
y + y^2 = x^2 + c
I am stuck trying to put in in explicit form, so y = ...
is this completing the square or something?
Homework Equations
The...
Homework Statement
Solve the separable differential equation for u
du/dt = e^(6u + 8t)
Use the following initial condition: u(0) = 13.
Homework Equations
Techniques for solving separable differential equations.
1. Group variable and respective dy,dx,dz, etc. together
2. Integrate both...
Homework Statement
Problem:
(1 pt) Let P(t) be the performance level of someone learning a skill as a function of the training time t. The derivative \displaystyle \frac{dP}{dt} represents the rate at which performance improves. If M is the maximum level of performance of which the...
Homework Statement
Find the y-intercept of the curve that passes through the point (4/9,1) and whose slope at (x,y) is −2/y3.
Homework Equations
The Attempt at a Solution
woudnt the y-int be y=-2x/y3 + 5.5 ?
i got it by plugging the slope into m in the equation y=mx+b
then...
(Moderator's note: thread moved from "Differential Equations")
Hi, I am trying to solve a differential equation which is almost separable, but not quite.
Specifically:
1 + \frac{dy}{dx} = f(x) g(y)
Is there any way to approach this, or are additional constraints needed? In the setting in...
Homework Statement
The following is an explanation from my tutorial. I do not understand it.
{\frac{d}{dx}(y_1{{y_2}'}-y_2{{y_1}'})+P(x)(y_1{{y_2}'}-y_2{{y_1}'})=0
Overlooking for the moment that P(x) may be undefined at certain values of x(so-called-singular points of the equation), we...
Homework Statement
Solve the following equation by separating variables.Homework Equations
x2y2y' +1 = yThe Attempt at a Solution
I have been able to work through the problem to this point:
-1/x + C = int(y2/y-1)dy
I am not sure how to integrate the right hand expression int(y2/y-1)dy
Homework Statement
The volume V of water in a particular container is related to the depth h of the water by the equation dV/dh below. If V = 0 when h = 0, find V when h = 4.Homework Equations
dV/dh = 16 sqrt(4-(h-2)2)
V(0) = 0
V(4) = ?The Attempt at a Solution
I have not gotten the ODE in a...
Homework Statement
Solve dy/dx = exp(x-y) given that y = ln 2 at x = 0Homework Equations
None.The Attempt at a Solution
Firstly let's get the equation into a form so we can re-arrange the x's and y's, and then re-arrange.
dy/dx = exp(x)/exp(y)
exp(y)*dy = exp(x)*dx
Integrate:
exp(y) =...
Hi,
I'm looking to show that the metric space of convergent complex sequences under the sup norm is not separable; that is what I assume it is since I cannot find a way to prove that it is separable (I am unable to find any dense subsets).
A set of complex sequences convergent to a certain...
Homework Statement
A tin organ pipe decays with age as a result of a chemical reaction that is catalyzed by the decayed tin. as a result, the rate at which the tin decays is proportional to the product of the amount of tin left and the amount that has already decayed. Let M be the total amount...
In Example 1.5 of Differential Equations Demystified, the equation is y' +2xy = x, and the author claims this is not separable. Now, what am I missing. I try the following.
1) dy/dx = x - 2xy
2) dy/dx = x(1-2y)
3) dy/(1-2y) = xdx
I guess one of those is invalid but I just...
Homework Statement
(x + 2y) dy/dx = 1, y(0) = 1
Homework Equations
The Attempt at a Solution
Problem is, I can't separate it. This might be a homogenous type? If so, how would I make it into the g(y/x) form.
Thank you.
Homework Statement
-y + xy' = 0 and y(2)=5
The Attempt at a Solution
This first part trips me up. I am supposed to find the perfect derivative, which is (xy') = 0 ? Is this legal, or does the -y not allow for that?
If that is correct, then I know that I integrate that, which...
So, here's the problem:
\frac{dy}{dx} = \frac{8y}{5x}
To start off, I separate the integrals, which gives me:
\frac{dy}{8y} = \frac{dy}{5x}
After that, I integrate both sides, which gives me:
\frac{ln8y}{8} = \frac{ln5x}{5} + c
Now, the question says that it runs through (4, 1), so that...
Homework Statement
\frac{dP}{dt}=P-P^{2}
It seems that Partial Fractions should be used to solve this D.E., but I cannot find an example to go by.
I even tried to rewrite the equation as:
\frac{d}{dx}Y\left(x\right)=Y\left(x\right)-Y\left(x\right)^{2}
But, that isn't helping me...
Homework Statement
\frac{dy}{dx}+2xy^{2}=0
I am stuck on this.
I realize that this is a non-linear exact equation, but I just cannot wrap my mind around any type of method to attack this one.
TIA for any help
I have tried to solve the differential equation
y'=x\sqrt{y}
like this:
y^{-\frac{1}{2}}y'=x
\int{y^{-\frac{1}{2}}}dy=\int{xdx}
y^{\frac{1}{2}}=\frac{x^2 +C}{4}
y=\left(\frac{x^2+C}{4}\right)^2
Is this the right way to solve it? Because the answer in my textbook says that the...