Math Is Fun Forum

Okay... so... I guess the key is to integrate by partial fractions because it gives a friendler form for answering the question.

You would have

Let

I do not remember how to integrate by partial fractions though. What are the steps for integrating the left side using partial fractions?

I did not think it mattered since C is whatever it ends up being in the long run.

Is that better? I just skipped to the end.

Hey...

Well, I hadn't integrated yet. I separated the variables and had the equality

and you solved the left side... and it's the same answer I was getting out of Maple, so thanks for verifying it!

Now this needs solved for A(t):

I assume it should just be solved algebraically. Here is my effort:

And so that is A(t). That is so complicated, though. Am I even moving in the right direction? What is the question really asking in relation to this?

Thanks for always hanging around to answer questions.

P. S. The solution of the ODE can be solved by hand. Here is how:

And solving for v(t) gives the general solution

Letting v(0) = 0 finds the particular solution

And you can factor out the g/k:

And so, in stating the answer as the original limit in terms of g and k,

In words: "The terminal velocity of a raindrop with mass m and velocity at time t is directly proportional to the acceleration due to gravity and inversely proportional to some constant k."

Neat! Now would this have been an easily-found solution by hand or no? I am still learning separation of variables and all that to do with natural logs, general solutions, and specific solutions.

The one you just solved in Mathematica is a good example.

In Maple I typed

to open the DETool box, then

In hopes of solving the differential equation. But I haven't really learned much about Maple's DE tools yet.

With Maple you can plot a couple of points more along the curve than just the three linear points, as I said, I was trying out the mean values and quarter and three-quarter lengths of the arcs, but the resultant curves were really bad. I figured I could either fit a ton of points or try to figure something else out more logical.

Fitting a thousand different points seemed like cheating... and it created gigantic functions, anyway. I was hoping the solution would be something simplistic and beautiful in that simplicity.

Exactly. A piecewise made into a single function h(x) or whatever we were calling it.

Part of my obsession with having one function instead of piecewise functions was experimentation I am doing in 3D surfaces which don't look so great when having to work with splines.

But, yes, a single function is the goal. One that matches the piecewise perfectly or at least extremely closely.

Correct - h(x), the unknown blue curve, is a piecewise function of f(x) from 0 to 75 and g(x) from 75 to 150. The ultimate goal is to fit a single curve defined by a single function.

The reason for using Thiele's Interpolation Formula was that it was a feature in Maple, yes. However, Maple also features polynomial, splines, linear, and so on but Thiele's formula does a perfect job on the regular curves shown in the majority of those pictures I uploaded.

The problem arises when the three primary points of the h(x) - the blue curve - are attempted to be fit for the points - (0,0), (75,25), and (150,50) - fit a perfectly linear line. But if more points from the original two functions f and g are chosen, say, the points of mean value, Thiele's formula does not fit very accurately if it fits at all for it is easy to hit vertical asymptotes as the formula divides by zero.

Polynomial interpolation does a terrible job of fitting the ideal h(x) perfectly.

I have read online about various interpolation methods but most of them are beyond me. One that at least sounded promising is found in this link:

http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula

However it is quite beyond me. Do you happen to know if that is an idea worth trying? I am willing to learn anything new, I am just out of ideas on my end.

I will be taking differential equations starting next month and I know the course has something to do with finding unknown functions. Since you have probably already taken such a course, do you happen to know if there is anything similar to this in said class?

Thanks a lot for getting back to me as quickly as you did.

bobbym,

1) Data? The two curves - f(x) and its inverse and on the stated boundaries - provide an infinite number of data points. Take your pick.

2)

Seeing as I have already performed an application of it, it should be fairly obvious "what I am trying to accomplish".

3.

My calculus class in junior college just happens to use Maple. It doesn't affect my amateurish status for good or not. Does it matter?

...

I apologize if my response is annoying, I simply didn't realize the question would be so difficult when it is the problem that has troubled me so. Perhaps it would help make the scenario, which is abstract in itself, easier for you if you imagine, say, a new road is being built that is half of an old function and half of the old function's inverse.

Perhaps I have simply made a mess of everything in attempting to explain it. An easier approach might be to simply start learning about new interpolation methods. I have tried reading about them online but it isn't the same as having someone explain it with the chance for questions and answers.

Also, I will try to get some images to support the question in the hopes that pictures will make the idea easier to understand. I will post again soon with said pictures.

Thanks...