Differential Equations

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


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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

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!

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.

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.

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.

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)?

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

quick help needed

y'' + 4y = -18 - 18cos(4t)
y(0)=y'(0)=0

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!