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24-Ruby II

## Has this equation one symbolic solution?

Has red equation one symbolic solution?

1 ACCEPTED SOLUTION

Accepted Solutions
12-Amethyst
(To:ValeryOchkov)
35 REPLIES 35
23-Emerald II
(To:ValeryOchkov)

Valery,

Try this.

Success!
Luc

24-Ruby IV
(To:ValeryOchkov)

You can easily calculate the integral by hand if you remember that 1+sinh^2(a) = cosh^2 (a). Then, using real arguments, the root "cancels" with the square and as X and arsinh(tan(alpha)) are constants, simple linear substiution gives you the integral. Then you can quickly solve for X and if yo do it manually you will arrive at the very same solution which Luc had provided.

But you can also let Mathcad's symbolics di the work. As usual when it gets a little more complicated Mathcad/Mupad needs a lot of help with substitution and the placement of the right commands in the correct order.

muPad seems to prefer ln instead of arsinh and asinh is not a valid argument for "rewrite", so it looks that there is no way to convince muPad to give the solution using arsinh instead of ln.

24-Ruby II
(To:Werner_E)

Thanks, Luc and Werner!

But If you are so smart, solve please this problem symbolically until the end!

24-Ruby IV
(To:ValeryOchkov)

@ValeryOchkov wrote:

Thanks, Luc and Werner!

But If you are so smart, solve please this problem symbolically until the end!

But if we do, there is no fun left for you ! 😉

Here's my attempt.

As you can see that in the last phase muPad switches to numeric mode. You may try your luck with something more capable like Maple or Mathematica, but chances are that there is not closed symbolic solution for that equation.

At least we see that the angle is constant and independent of the parameter L and the symbolics arrives at the same result as the numeric "root" function. At least sometimes. Sometimes, as I show, muPad would return results which are simply wrong! Sure a no-go - this should never happen!.

Sheet in MC11 format attached - maybe Luc is about to give it a try with Maple.

23-Emerald II
(To:ValeryOchkov)

Here's how far I get:

Check if these results correspond to the solve block results:

OK, now we can continue:

So here it ends.

Luc

24-Ruby II
(To:LucMeekes)

Today I saw such a symbolic solution of the problem in my dream.

23-Emerald II
(To:ValeryOchkov)

Note that 7pi/25 is far off from the solution.

Luc

24-Ruby II
(To:LucMeekes)

23-Emerald II
(To:ValeryOchkov)

Nice approximation. Had it been exact, I'd have gotten it from WolframAlpha, I guess.

You cannot calculate every real number (from the R set of numbers) from the quotient of of two integer numbers (that is, by a number from set Q). I suppose (but just that) that it also holds true if you multiply the quotient with a real number such as pi...

Luc

24-Ruby IV
(To:ValeryOchkov)

@ValeryOchkov wrote:

A better approximation still remains an approximation and is sure no symbolic solution.

I don't understand why you think that the solution would be a rational multiple of pi and why you are playing around the way you do.

You already got a numeric approximation using root and I showed how to get a (probably more precise) numeric approximation using Mathcad's symbolics.

What would be the benefit of writing those approximations as fractions and multiples of pi???

You sure can increase the number of decimals (use the symbolics "float,100") to get even more precise approximations but they remain approximations and i see no sense in doing so.

12-Amethyst
(To:ValeryOchkov)

12-Amethyst
(To:ValeryOchkov)

24-Ruby II
(To:mnori)

I think it is one new Physical and Math constant!

Steven Finch! Where are you?

24-Ruby II
(To:ValeryOchkov)

More Physic less Math - a numerical solution of the system of 3 physical equations!

15-Moonstone
(To:ValeryOchkov)

Just for fun I thought I'd see how far I could get on Valery's problem with a combination of Mathcad symbolics plus "Hand" symbolics.  Like others, I found it's easy (if tedious) until trying to find where PE = -L^2/4.  At this point one needs to resort to numerics.  Here's my approach (in MC15):

Alan

24-Ruby II
(To:AlanStevens)

Thanks, Alan!

But solution without animation is not solution

23-Emerald I
(To:ValeryOchkov)

@ValeryOchkov wrote:

Thanks, Alan!

But solution without animation is not solution

Disagree!  Alan did the hard work, the animation is window dressing.

24-Ruby II
(To:Fred_Kohlhepp)

I came up with not only a problem, but a new joke (an anecdote):
"One Italian, one Austrian, one American, one Englishman and one Dutch met together someone and began to solve one problem from Valery..."

Laughter laughs, but I want to publish this problem in one serious journal.
It is very important for me. Help me please  with the language and the rest!

24-Ruby II
(To:AlanStevens)

@AlanStevens wrote:

Just for fun I thought I'd see how far I could get on Valery's problem with a combination of Mathcad symbolics plus "Hand" symbolics.

Alan

"Numeric + Symbolic + "hand" symbolic = hybrid calculation" - it is a name of one chapter of my new book. Now it is in Russian (see please here). But I have a plan to publish it in English too.

Alan! Check please the translation of this chapter (study) in attach. Remarks will be good too not only from Alan bot from Community!

15-Moonstone
(To:ValeryOchkov)

I don't really have the time to go through your document fully, Valery.  However, I spotted a few word changes needed in the first paragraph:

"etude" should be "study".  ("etude" is French)

"analog" should be "analogue"

"architecture" should be "architectural"

"refusal" should be "decline"

Good luck with the book!

Alan

24-Ruby II
(To:ValeryOchkov)

Ease to remember this constant: 50+20+20=90

12-Amethyst
(To:ValeryOchkov)

I made some mathematical tourism, I read that this is Gudermannian function, here there is a discussion with several links:

24-Ruby II
(To:mnori)

We have created the MOSNE function

(Meekes, Ochkov, Stevens, Nori and Exinger)

24-Ruby II
(To:ValeryOchkov)

23-Emerald I
(To:ValeryOchkov)

That's impressive!

It would be REALLY impressive if it were Mathcad!

24-Ruby II
(To:Fred_Kohlhepp)

@Fred_Kohlhepp wrote:

That's impressive!

It would be REALLY impressive if it were Mathcad!

23-Emerald I
(To:ValeryOchkov)

@ValeryOchkov wrote:

@Fred_Kohlhepp wrote:

That's impressive!

It would be REALLY impressive if it were Mathcad!

I meant the image, not the mathematics.

In college, the physics course in electricity and magnetism included a laboratory.  The professor was demonstrating the use of an oscilloscope during the lab.  He said if we could design an experiment that would write our names using the oscilloscope he would give us top marks in the course and we could stop attending.

24-Ruby II
(To:Fred_Kohlhepp)

@Fred_Kohlhepp wrote:

@ValeryOchkov wrote:

@Fred_Kohlhepp wrote:

It is MathcadIn college, the physics course in electricity and magnetism included a laboratory.  The professor was demonstrating the use of an oscilloscope during the lab.  He said if we could design an experiment that would write our names using the oscilloscope he would give us top marks in the course and we could stop attending.

Something like this

24-Ruby II
(To:ValeryOchkov)

But the High Tech for the names and others drawing is not electrical but mechanical device!

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