Has red equation one symbolic solution?
See please the picture and the Mathcad 15 sheet in attach!
Solved! Go to 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.
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.
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...
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???
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.
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):
But solution without animation is not solution
Disagree! Alan did the hard work, the animation is window dressing.
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!
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.
"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!
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!
I made some mathematical tourism, I read that this is Gudermannian function, here there is a discussion with several links:
It would be REALLY impressive if it were Mathcad!
It is 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.
If your image was a Mathcad plot, please post the file.
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