I have been trying to understand the difference between linear and nonlinear differential equations. If one has some random differential equation of the form
then, if I am correct, it is first order because it is the first derivative and ordinary because there is only one independent variable, being x, correct?
Now suppose the answer is known to be some rational expression of the form
where P(x) and Q(x) are polynomials, and let's say they are of the first degree, meaning the highest order of x is to the first power.
Is the original differential equation linear or nonlinear? The result is not a linear function and has nothing to do with transcendentals so far as I am aware. I would think that the original equation would be called nonlinear but I wanted to make sure.
Hello. I have a definition question.
According to wikipedia and wolfram alpha, impredicativity is the self-reference of a set. What would be a simple example of this?
If we have a point on (x,f(x)), say, (a,b) and separate the two points as
so that either self-referencing result is not a function but is, at the specified value, the value itself... would this be a good example of the concept? Each result has x or f on both sides of the equals sign so that if x=a and f=b then we revert to our original values that constitute a single point.
Hi!. Long time no post.
I have a question about finding the vertex of a hyperbola in elliptic coordinates. Suppose your coordinate system is given by
and suppose, somehow, you happen to know that a≠1 and you also happen to know a few of the points along the hyperbola... I don't know, maybe you have a diagram or something that shows the points labeled.
Does anyone know of a method for finding a≠1 based entirely on a few points along the hyperbola?
I did it but it's more typing than I have time for right now. You have to multiply M by a matrix with 9 unknowns so that you end up with a system of nine equations with 11 unknowns and then solve it for those unknowns entirely in terms of a and b only. It was quite a bit of substitution and algebraic manipulation and the resulting transformation matrix is enormous.
Thanks for your input!
Quite possible. I was just trying to simplify the model for the sake of those reading it but I may have messed it up in doing so.
If simplification complicated the issue, the function f(x) might even be able to expand with some manipulation. For instance letting c=a-b or something to eliminate c while adding in additional a and b terms. Or not. Actually, that might make it worse.
The problem I introduced stemmed from experimentation with representing that hyperbolic function I keep bringing up in matrix form. As bob bundy implied, the function
may be rewritten as
which, according to wikipedia, is of the form
where 2B=1, 2D=-a/c, and 2E=b/c, and everything else is zero. Therefore we can state the original f(x)=y in matrix form as
In coming here I just used Greek letters to make the problem more simple rather than using fractions for every term. I left the 1/2's in case they held significance.
So then the problem became finding a matrix P that would transform a more basic equation y=x into the equation f(x) given at the start. The function y=x has the matrix
but that didn't work at all which is why I added in the two extra 1/2's in to M because if those terms were left as 0s then no matter what I multiplied by it, I would still get 0. And so, essentially, my original question was to ask for a transformation matrix P that would transform R in to M; that is, to go from
using matrices and a matrix transformation.
How does one go about transforming an arbitrary matrix, say, one of the form
to the form
where the only change are the Greek components?
I tried just working out a matrix P on the spot to multiply M by to get N but I didn't get what I wanted at all. I have had a little experience with matrix transformations but I don't know how to derive the transformation itself. If someone could give me the matrix and explain how you got it so I understand, I would appreciate it.
Thank you for your help.
We can form a second order differential equation and get it from there.
bobbym: Which notation of the initial statement would you say is the most correct?
or something else? They can all be interpreted to mean the same thing but I do not know if any of them are better than the rest.
I see what you are saying and I already know that. When it is around zero it is a Maclaurin series. My original question is, how do I go from
without the aid of a computer program like Maple? Is there a way to compute it by hand like an integral? That is the information I am lacking.
I am trying to teach myself infinite series. I recently posted on this forum questions involving the relationships between rational expressions and hyperbolas and because I am still in that mindset, I am using that example.
I have managed to work out that the series expansion or infinite series or infinite sum or whatever it is called for a function
will be of the form
if computed will yield the desired function. I have confirmed this in Maple.
How do I go about doing this by hand so as to show all my work? Computing it in Maple is easy but I want to know how to do it for myself. I attempted the problem myself, treating it as an integration problem but that didn't work and it makes sense that it didn't work because I am counting to infinity instead of finding the area under a curve.
Any help is appreciated.
Okay, here is what I have. I would appreciate any edits or suggestions anyone who knows sets or mathematical writings might have to offer to help improve the quality of what I have so far.
If time is initially defined as a well-ordered uncountable set
[in that it might be said to go from ±∞], the set T might be written so as to contain some boundary condition
so that it can be rewritten as;
but if we further define the set as being of certain relevance to only that boundary condition Z, we say let
so as to be redefined as.
Suppose then we have a series of events in time that are also restricted to Z (which is actually what I intend to be the reason for why Z is a boundary condition in the first place) and let that ordered set (ordered because it corresponds to time T) be given by.
Because every event in E corresponds to no more and no less than one event in T,
(meant to read: a function t is the mapping of the set E to the set T such that for all t's in the set T there exists one and only one e in E such that t as a function of e is equal to t)
such that for a bijector J between T and E,
that is, the cardinality of both sets is equal such that the magnitude of both sets is equal:.
Thank you for any and all input. Any and all critiques and edits are welcome.
P. S. Yes, starting out with the most fundamental definition of time is a goal, then to be worked out so as to be limited to the specified boundary condition which is so-bounded because of the set of E being the only set where T is relevant.