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Differential Equations

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Defective pixel [03 Jan 2010|05:44pm]

ddt79
There is true expression on panel. But only one pixel is defective. Which one?


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[18 May 2009|03:48pm]

eck_lesi
A general question. I am studying a certain system that seems to generate oscillations whose phase is coupled with the amplitude. The exact form of coupling is unknown at this point, so this is what I am trying to study. I am sure that similar effects must be known in many fields and that there must be ways of studying such waves, but for me it's something new. I was wondering, can someone suggest some readable literature on this topic, something that would help me to build intuition about it?

May be some empirical tools (time series analysis? others?) for studying empirical data of this sort? Some applied theory behind such phenomena?

Thanks
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quasilinear PDE help request [18 Sep 2008|08:59am]

quasigeostrophy
The problem:

Solve yux - xuy = 0 , y > 0 with the condition u = y when x2 + 2y2 = 4. What is the maximal domain in y > 0 where the solution is valid and determined uniquely by its Cauchy data? Draw a sketch.

What I've done so far:

If I'm correct, I've gotten the initial set of ODEs as dy/y = dx/-x and dz/dt = 0, hopefully leading to x2 + y2 = s and z = s, and therefore u = f(x2 + y2).

I know that u = y is a tilted plane that will intersect the surface of revolution of x2 + 2y2 = 4 in a half-ellipse (y > 0). I'm not sure how to finish or answer the second part of the question, or if I'm right so far.

Thanks much in advance for any hints or direction.
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An Initial Value Problem [16 Sep 2007|07:27pm]
rscholar

Could you help me to solve an initial value problem:

sin 2y dx = (sin^2 2y - 2 sin^2 y + 2x) dy, y(-1/2)=pi/4

Thank you in advance!

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differential equation theory [27 Mar 2007|09:14am]

ibid14
[ mood | curious ]

Just a measly math-minor freshman here...

I understand that the linear combination of two solutions is itself a solution to homogeneous differential equations, but what I don't understand is why we only use two.  Why stop there?  Why not use the linear combination of three equations?  Or four?  It seems like if you're going to say one is not sufficient, but two is, you have to specify why even more is unnecessary.  Does this make sense?

Granted, my school's (UNC-Chapel Hill) math department isn't its greatest department, and my diff eq teacher doesn't really spend much time teaching us the theory behind everything--just how to solve them, so this may actually be a regularly-covered topic.  But I don't remember it from my high school diff eq class either.

Maybe mathematicians are just lazy?



EDIT

"Problem" solved.  Contribute more if you really want to, though.

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[21 Jan 2007|08:43pm]

viccro
[ mood | frustrated ]

Hey!
I think I meant to say 'Help!'

I can't figure out how to "write a 1st order linear DE for which all solutions are asymptotic to the line y=3-t as t->inf."

Then I have to do the same for y=4-t^2.

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hi there, i need some help [25 Aug 2006|07:47pm]

ex_exzelsior746
my problem is to solve a set of Einstein equation numerically. the method is simple - by the matrix representation of hamiltonian. but i've faced an internal technical problem:

Error, (in evalf/EigensRG) both matrices must have the same dimension

here is the maple code:

> restart;
> L1:=-(1/2)*diff(psi(a,n),a,a)+(1/2)*(a^2)*psi(a,n)=E*psi(a,n);
> H:=(-1/2)*Diff( Psi(a,n),x$2)+V*Psi(a,n);
> V:=(1/2)*a^2;
> with(orthopoly,L);
> L(3,a);
> f:=exp(-a/2);
> psi:=unapply(f*L(n,a),a,n);
> Hmn:=(m,n)->Int(psi(a,m)*((-1/2)*diff(psi(a,n),a$2)+V*psi(a,n)), a=0..infinity);
> with(linalg):
> p:=1:
for k from 5 by 1 while abs(p)>0.1 do
H1:=array(1..k,1..k):
for i from 1 to k do
for j from 1 to k do
H1[i,j]:=value(Hmn(i,j)):od:od:
H2:=array(1..k+1,1..k+1):
for i from 1 to k+1 do
for j from 1 to k+1 do
H2[i,j]:=value(Hmn(i,j)):od:od:
E1_i:=convert(evalf(Eigenvals(H1,'vects1')),list):
E1_s:=sort(E1_i,numeric):
E2_i:=convert(evalf(Eigenvals(H2,'vects2')),list):
E2_s:=sort(E2_i,numeric):
p:=E2_s[1]-E1_s[1]:
p:=E2_s[2]-E1_s[2]:
p:=E2_s[3]-E1_s[3]:
od:
k;


Error, (in evalf/EigensRG) both matrices must have the same dimension

what can be a treatment?
it should be a simple thing, i suppose.
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[05 Mar 2006|08:06pm]

wetpanda
so, i posted this not too long ago in the mathematics community, but didn't receive much of a response. so, i now post in here, where it seems perhaps more at home:

what is the general solution to ∂f/∂x = f(x, y-1)?
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Analysing a system by way of eigenvalues [27 Feb 2006|05:13pm]

shamebear
I'm analysing a physical system to say something qualitatively about its stability and likely future trajectory in phase-space. It's a thermodynamically open one, with frequent interference by humans. I'm brainstorming for ways to do this and would appreciate your feedback. :-)

I have a rough but incomplete model of the causal pathways and plenty of historical data. If this had been a linear system, I could have done the following: Set up the Jacobian matrix (containing all partial derivatives of all observable relative to all others) from the following formula:
X' = J*X
where J is the jacobian matrix and X is a vector with each observable x1(x2,..,xn) to XN(x1,...x(n-1))

If I look at the eigenvalues of the Jacobian, their values can tell me about the qualitative behaviour of the system. Exponential growth, oscillation or convergence toward an attractor. (But I think I can not employ Lyapunov stability as it's not an undisturbed or closed system) In system dynamics, a technique called eigenvalue elasticity analysis uses this information to point out the dominant feedback loops in the system. (Which would be really useful)

The real physical system is probably nonlinear and my observables are probably only a subset of the variables that form the "real" causal chains. I'm not in a position to experiment on the real system, but I could experiment on a computer model, if only to reveal its shortcomings.

Extending the eigenvalue analysis to nonlinear systems is apparently a big topic (yes I googled :) and I'm not sure where to begin. Things I want to accomplish is:

* Sketch the phase-space reachable in the next few hours with or without human intervention, taking into account noisy real-time measurements
* Do an analysis of the computer model to document where there might be missing causal chains.
* Find what feedback loops are dominating the systems' behaviour, perhaps to have the computer model do more fine-grained analysis of these loops.
* Reduce the dimensionality of the problem. Perhaps not the general problem, but the time-evolution from time t1 to t2 from a given point in phase space.

The end goal is to avoid unwanted behaviour of the system, pick up warning signs in advance and present simplified causal chains to humans on a case-by-case basis. Do you have any leads or warnings in relation to this?

crossposted to mathematics
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[16 Dec 2005|09:08am]

emoboygenius
quick help needed

y'' + 4y = -18 - 18cos(4t)
y(0)=y'(0)=0
2 comments|post comment

I love the engineering hall [18 Nov 2005|02:09pm]

spaceyphysicist
[ mood | cold ]

So last night I was going through freshman hall that's all aerospace engineering majors, and there was something written on the whiteboard in the lounge that amused me to no end.

NASA kills monkeys.
NASA uses Matlab.
Therefore, Matlab kills monkeys.

Thought you guys would appreciate it. ;)

viva mathematica!

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pde prob [16 Nov 2005|09:09pm]

emoboygenius
Two things
same related question:

The Setup:
U_tt - (c^2)U_xx = -rU_t
resistance r > 0 and constant
periodic source U(0,t) = U(L,t) = Ae^(iwt)

Questions:
first says show that you can get resonance as r -> 0 if w = m(pi)c/L
second show that friction can prevent resonance from occuring

Our teacher never went over this in class, and this is the only time it's mentioned in the book so yeah I don't really know what they are talking about
any ideas?

oh and it gives the possible solution for the PDE as U(x,t) = Ae^(iwt) sin(bx)/sin(bL) where b^2 c^2 = w^2 - irw

x-posted
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Help! [11 Nov 2005|03:42pm]

mathphysicsgeek
I'm feeling really pathetic right now... I can't remember how to solve the following equation:

y'(t) = 16*y(t)*(1-y(t))
y(0) = 1/1024

Can anyone help me with that? Thanks!
5 comments|post comment

Boyce & DiPrima's Elementary Differential Equations [27 Oct 2005|11:38am]

clearheaven
[ mood | excited ]

Hello,

I don't know if this is allowed in the community, but I have stumbled across an interesting discovery online. It is the 7th edition of Boyce & DiPrima's Elementary Differential Equations, the same textbook I used for my DiffEQ course. I would like to share it with you in hopes that it may be useful to you.

http://ephemeraleuphoria.com/media/mathematics.diffeq.boyce-diprima.7thEd.pdf

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Dynamics probs [14 Oct 2005|04:49pm]

emoboygenius
I had a test wednesday and I'm still trying to figure out if i got one of the questions right
it asked to turn a dimensional problem dimensionless
dx/dt = rx -x^3 + x^5

After a lot of screwing with the problem I showed two ways in which I felt it was dimensionless
first:
[x'] = [rx] = [x^3] = [x^5]
this means that x is dimensionless b/c that's the only way [x^3] = [x^5], which also means that r has to be dimensionless
which leads to the entire right hand side, which means the problem is
second:
expanding the right hand side you get
-x(x-(1+(1+4r)^1/2)/2)(x-(1-(1+4r)^1/2)/2)
if you look at one of the parenthesis you see x - .5 + ((1+4r)^1/2))/2
using simple logic you know to add these [x] = [.5]
since .5 is dimensionless x is dimensionless
which leads back to the whole line of thought that if x is dimensionless the entire problem is

now, I wouldn't be asking this if I hadnt talked to my prof today and showed him what I did and he looked at me kinda weird and said hmmmm i'll have to look into that and walked off
that and that the left hand side has a dt on it
oi
so yeah, confusion abounds
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Help! Linear Algebra [03 Oct 2005|06:25pm]

exzebachay
We're covering Images and Kernels in my Diff Eq class, and I'm having some problems with images of linear transformations.

From our book:
"Suppose that L:R^m --> R^n is a linear transformation. The vector space R^m is called the domain and the vector space R^n is called the range of L. This terminology is just the standard terminology that one uses when working with functions. Notice that if v is a vector in the range of f that there is no guarantee that there is a vector u in the domain of f for which f(u)=v. For example if L:(x,y) --> (x+y+x, 2x, 0) is a linear transformation from R^2 to R^3, then there is no vector u in R^2 for which f(u)=(0,0,1). The phenomenon also occurs when studying functions of a single variable; for example if f(x)=x^2, then there is no x=a for which f(a)=-2. The definition below makes a distinction between a set that merely contains all possible outputs of a function and the set that actually equals all of these outputs."

What? It specifically states that "v is a vector in the range of f", so how can we even consider that for f(x)=x^2, then there is no x=a for which f(a)=-2? -2 isn't in the range of f...
The book defines image(L) as being a single set that's contained within range(L), but I have no idea how to calculate it.

Can someone give me some pointers?
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[28 Sep 2005|11:52pm]

emoboygenius
hey i havent taken pde or ode in a hella long time so i'm a bit rough and now im stuck on a very seemingly simple problem and need some help
general solution of 3U_y + U_xy = 0
now I found that if you make V = U_y then you get 3V + V_x = 0

from there you can do a integral of the partial and get V = e^(-3x)e^(f(y)) = U_y
that's where I get stuck
I know it's something stupid, but I've been working on pdes for a while and my brains farting
any help would be appreciated
btw is there a pde or dynamics comm?
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Tangential Question About TI-89 [09 Sep 2005|08:12pm]

warrocketajax
Sorry for being off-topic, but there's a startling lack of forums about calculator programming and I figured this would be a good place to check.

Can anybody tell me if a TI-89 function can be passed mathematical functions as an argument? For instance, I'm trying to write a series of programs that automate numerical methods like bisection method, Newton-Raphson, etc. I would really like to enter the analyzed functions from the command line as parameters instead of having to edit functions every time I start a new problem. The input I'm looking for would be something like this:

bisect(f,x,y,e)

Where
f= f(x)
x= beginning of interval
y= end of interval
e= prescribed error tolerance

I'd appreciate any help. Thanks in advance.
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[25 Jul 2005|09:33pm]

erlc
I know this isn't really on topic, so feel free to delete it if it hurts your feelings or something.

does anyone have the solutions manuals for:

Fundamentals of Thermodynamics, 6th edition - by Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen

Engineering Mechanics - Statics, 10th edition - by Russell C. Hibbeler


thanks^_^

- another fellow engineering student
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Free copy of DiffEQ textbook by Boyce & DiPrima [29 Jun 2005|02:28pm]

clearheaven
Hello DiffEQ students (AKA tortured souls),

I have the full solutions manual in PDF format of a popular differential equations textbook:

Boyce, William & DiPrima, Richard. Elementary Differential Equations and Boundary Value Problems. 8th Ed. 2004.

It is currently being sold on Ebay and half.com for about $6.00. Since I already have it, I would llike to share so as to not rip off my friends and help as many as I can. If you would like a copy of it for free, simply e-mail me at the address noted in my user info. I would be more than happy to help you.

-A fellow engineering student
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