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Has this equation one symbolic solution?

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Emerald IV

Has this equation one symbolic solution?

Has red equation one symbolic solution?

See please the picture and the Mathcad 15 sheet in attach!

Closed-Chain-Alpha-PE-Root-numeric.png

1 ACCEPTED SOLUTION

Accepted Solutions
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Re: Has this equation one symbolic solution?

35 REPLIES 35
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Re: Has this equation one symbolic solution?

Valery,

Try this.

LM_20180806_SymbolicSolution.png

Success!
Luc

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Re: Has this equation one symbolic solution?

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.

B.png

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Re: Has this equation one symbolic solution?

Thanks, Luc and Werner!

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

Closed-Chain-Alpha-PE-symbolic.png

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Re: Has this equation one symbolic solution?


@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!.

B.png

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

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Re: Has this equation one symbolic solution?

Here's how far I get:

LM_20180806_NumericChain.png

LM_20180806_SymbolicChain1.png

Check if these results correspond to the solve block results:

LM_20180806_SymbolicChain2.png

OK, now we can continue:

LM_20180806_SymbolicChain3.png

LM_20180806_SymbolicChain4.png

So here it ends.

 

Luc

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Re: Has this equation one symbolic solution?

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

GuessValueSymbolic.png

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Re: Has this equation one symbolic solution?

In your dreams! (only).

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

 

Luc

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Re: Has this equation one symbolic solution?

and what about this "solution"

GuessValueSymbolic-2.png

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Re: Has this equation one symbolic solution?

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

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Re: Has this equation one symbolic solution?


@ValeryOchkov wrote:

and what about this "solution"

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???

B.png

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.

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Re: Has this equation one symbolic solution?

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Re: Has this equation one symbolic solution?

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Re: Has this equation one symbolic solution?

I think it is one new Physical and Math constant!

Steven Finch! Where are you?

Closed-Chain-Alpha-close-formula.png

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Re: Has this equation one symbolic solution?

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

Fig-16.png

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Re: Has this equation one symbolic solution?

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):

 

Valery1.PNG

Valery2.PNG

Valery3.PNG

Valery4.PNG

Alan

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Re: Has this equation one symbolic solution?

Thanks, Alan!

But solution without animation is not solution

Closed-Chain-Alpha-close-formula.gif

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Re: Has this equation one symbolic solution?


@ValeryOchkov wrote:

Thanks, Alan!

But solution without animation is not solution

Closed-Chain-Alpha-close-formula.gif


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

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Re: Has this equation one symbolic solution?

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!

 

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Re: Has this equation one symbolic solution?


@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!

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Re: Has this equation one symbolic solution?

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

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Re: Has this equation one symbolic solution?

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

Fig-16-90.png

 

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Re: Has this equation one symbolic solution?

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

http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1993229&messageID=6864360

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Re: Has this equation one symbolic solution?

We have created the MOSNE function

(Meekes, Ochkov, Stevens, Nori and Exinger)

MOSNE.png

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Re: Has this equation one symbolic solution?

handle-formula.png

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Re: Has this equation one symbolic solution?

That's impressive!

 

It would be REALLY impressive if it were Mathcad!  Smiley Happy

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Re: Has this equation one symbolic solution?


@FredKohlhepp wrote:

That's impressive!

 

It would be REALLY impressive if it were Mathcad!  Smiley Happy


It is Mathcad

handle-formula-Mathcad.png

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Re: Has this equation one symbolic solution?


@ValeryOchkov wrote:

@FredKohlhepp wrote:

That's impressive!

 

It would be REALLY impressive if it were Mathcad!  Smiley Happy


It is Mathcad

handle-formula-Mathcad.png


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.

 

If your image was a Mathcad plot, please post the file.  Smiley Happy

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Re: Has this equation one symbolic solution?


@FredKohlhepp wrote:

@ValeryOchkov wrote:

@FredKohlhepp 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

oscil.png

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Re: Has this equation one symbolic solution?

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

4-Circle-8-2.gif

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