Math Is Fun Forum

Hello.

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.

Thanks.

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

and

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.

Thanks.

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?

Thanks.

Hello.

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

such that

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.