I am trying to find a dynamical system with two differential equations, x(dot) and y(dot), that are in terms of x,y,or both, where regardless of where the system starts it will go towards the origin eventually make a counter-clockwise circle around it-- I have tried manipulating the circle equation, but always have dx/dt and dy/dt in both equations, which I can't get out-- Any help?

"it will go towards the origin eventually make a counter-clockwise circle around it"

What do you mean? Does it go towards the origin or does it make a counter-clockwise circle around the origin?

If it's just making a counterclockwise circle around the origin it's pretty simple. Just think of how you need dx/dt and dy/dt to be at the points (1,0), (-1,0), (0,1) and (0,-1).

Well it has to go towards the origin and then make the counter clockwise circle around the origin-- So no matter where it starts, i.e. (500,200) it has to go towards the origin and then make the circle

I don't think it will ever actually make a circle, but dx/dt=-y/100 and dy/dt=x/100 would move toward the origin circling it counterclockwise. I don't think you can have every point move toward a stable orbit of a circle of radius greater than 0 anyway, since points inside the circle would move away from the origin onto that circle. I could be wrong about that, though, I don't really know that much about dynamical systems.

Alright thanks-- I understand the circle now, I just wasn't thinking about it in the right way-- Last ? -- I need to find a dynamical system that converges on the origin from any point and will display exponentially increasing speed, eventually crashing into the origin as the speed approaches infinity-- I figured out how to have it work if speed is to remain constant (-x or y/ sqrt(x^2+y^2), but when I try to make the speed increase exponentially I can't get it to work from all points-- Thanks again for the help

What if you use dx/dt=-x/(x^2+y^2)?

I am having trouble figuring out a few derivative type problems if anyone could help it would be greatly appreciated.


The first I need to find the rate of change. The Formula given is 100t/(t^2+20t+100).
I am not sure what formula I use to find the derivative of that formula. I tried using the quotient rule and I got what I think might be the right answer but I am unsure.

The other problem gives me a cost function 5x^2+2 and it asks me for the marginal cost function, derivative of the average cost function. If anyone could help me figure out the derivatives of either two functions I would appreciate it.

Quotient rule works for the first function.

answer is [(t^2 + 20t+100)(100) - (100t)(2t+20)]/[(t^2+20t+100)^2]

The second one just take the derivative to get MC. (10x)

Average cost = TC/x

= 5x+2/x

take the derivative you get 5-2/x^2

5.3.13
Suppose a coin is to be tossed n times for purposes of estimating p, where p =P(heads). How large must n be to gurantee that the length of the 99% Confidence Interval for p will be less than .002?

I used the formula:

n=(z*/m)^2(p*(1-p*)

n=(2.576/.02)^2*(.5*.5)=4147.36

the book is giving me the answer 16641

that should read "confidence interval is less than .02"

I really wanted to post this, but I wasnt going to make a new thread. Taken from IRC via QDB.

<charl> hey navi, help. 9x - 7i > 3(3x-7u), for i
<navi> i <3 u
<navi> .. i'm going to kill you in your sleep

ok, we've been assigned the derivation for finding the angle of Rutherford's scattering experiment.





i have spent hours trying to do this but i simply cannot get the psi integral to work out to what they give as psi. i have worked out everything before and after that and have seemingly gotten painfully close but just cannot get the answer

the goal is to get to the psi equation in the second image

nevermind the above post,we got it, stupid trig tricks

Hi, I'm only doing As (UK) level maths and am stumped by this question:

A cuboid has a square base of length x cm and height y cm.
Volume = x^2y
Surface area = 2x^2 + 4xy

Show that V = 6x - 0.5x^3

I think I have to find y and substitute but struggling with this step.

Any help appreciated, cheers.

Solved it.

So I have a homework problem that I am not really understanding what the teacher is looking for-- Basically we are studying a pendulum with no friction, no air resistance and examining the associated dynamical system-- I was able to derive the basic dynamical equations, namely (dx/dt)=y and (dy/dt)= -sin(x) (we are not using the earth constant for g, hence the negative sign)-- After this, our teacher said there were a couple very unique trajectories in the phase plane that has the dynamical system (x-axis = theta, y axis= d(theta)/dt)-- I simply discussed the two types trajectories, the ellipse and the trajectory that goes on for infinity, but he said he was looking for two very specific ones-- Any idea what these might be?

Nevermind, I think I figured out what he was looking for

This should be a simple proof, but I'm just blanking on it:
Let A,Q be in R^(nxn), let Q be invertible, and for some norm |.|, let

|I - inv(Q)*A| < 1

Prove that A is invertible.

I think this should be a really quick proof (since this is part 1 to a multi-part problem, and the rest of the parts are 2-3 line proofs).

I thought about splitting it up into cases. If the norm=0, then A=Q, and its invertible.
Otherwise pick a vector x in R^n that maximizes
|(I-inv(Q)*A)x|
This value is <|x|.

Now what?

This is a trick you should remember. You can write

A^{-1} = (I-(I-Q^{-1}A))^{-1}Q^{-1}.

Now, if these were real numbers, we would have (1-x)^{-1} = 1+x+x^2+...

So you need to prove that (I-(I-Q^{-1}A))^{-1} = \sum_{n\geq 0} (I-Q^{-1}A)^n. This sum converges, basically since ||I-Q^{-1}A||<1, but you should justify that it actually is the inverse of what it's supposed to be the inverse of, as well as all the other steps.

I don't know what theorems you are allowed to use, but there's a well-known result that says that if |A-B|<1/|A^-1|, where A is an invertible matrix, that B is also invertible. It's in rudin, e.g. I might be slightly misquoting the theorem, but I'm pretty sure that's it.

Is anyone familiar with the notation u ∈ C²₀(U), where u:Rⁿ->R and U is a bounded smooth subset of Rⁿ?

Presumably the '2' in the superscript just means u is twice continuously differentiable; but I haven't seen a '0' subscript before.

C^2_0(U) is the twice continuously differentiable functions whose support is compact (the support of a function f is the closure of the set on which it is nonzero)

What is the drag force on a 1.6 m wide, 1.4 m high car traveling at 10 m/s?

anyone know how to take limits of recursive functions? i have the recursion a_n+1=a_n+6a_n-1 with conditions a_0=0 and a_1=1 and i need to find the limit for n going to infinity of a_n+1/a_n. i know from the generating function that the limit is supposed to be 3 but i can't figure out how to prove it with a limit.