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Wasi Husain
Apr 27, 2021

Help for a hyperbolic trig equation

in Math Problems

Can some one help me to solve the following equation for x :



It actually popped out of a physics problem that I was solving and is still not solved.

7 answers5 replies
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1
Faizan Ahmed
Apr 27, 2021

drive.google.com
problem.pdf

drive.google.com
problem.docx

Ian Fowler
Apr 27, 2021  ·  Edited: Apr 27, 2021

Sorry Faizan. In your solution, 2y = ln[sqrt(m) +t] / [sqrt(m) + t] leads to 2y = ln(1) = 0

So y = 0 leads to 2t - 1 = 0 or t = 1/2

That leads to x = [1-4m]/4pA.


But it's worse than that because I think you have made a sign error in line 13:

e^2y = ln { [sqrt(m) + t] / [sqrt(m) - t)] } - the green "-"

You will not be able to isolate t. Too bad about the minus sign. Otherwise it would have worked out.

Wasi: tanh[f(y)] = g(y) cannot be solved for y unless f(y) is a constant. Maybe there is some kind of convoluted Lambert function solution.


That's my take, unless I have missed something obvious.

0
Wasi Husain
Apr 28, 2021

@Ian Fowler Any solution will be ok even if it uses either lambert W function or some non - elementary function. Until the solution is computable it is okay for me.

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1
Faizan Ahmed
Apr 29, 2021

@Ian Fowler

Thanks for pointing out the typo error The updated solution is here.

Solution
.docx
Download DOCX • 23KB

Solution
.pdf
Download PDF • 80KB

1
Ian Fowler
Apr 29, 2021

I hate to be the fly in the ointment again, but here's throwing caution to the wind.

In line 15 you have some of the form : e^2y = B.

In log form this become: 2y = ln(B),

not 2y = B which what you have in line 17


You can even take the route of e^2y = e^ln(B) , which is unnecessary, but it still leads to 2y = ln(B). So line 17 is missing the ln function.


The bottom line is you have 2y = ln( a function of y) which does not have an elementary solution. Maybe there is some kind of Lambert function solution but that's the best you have.


Wasi: You might try using the infinite Maclauren series for arc_tanh(x) and use a computer to grind out some desired level of accuracy and then go from there or the Lambert function possibility. I think those 2 are your only shot.

1
Ian Fowler
Apr 30, 2021  ·  Edited: Apr 30, 2021

Faizan: I don't mean to flog the point but you missed the 2nd part of my response.

In line 15 you have: e^2y = B where B = [sqrt(m) +t] / [sqrt(m) - t]

Then on line 17 you have : 2y = B

This is incorrect as line 15 implies that: 2y = ln(B)

So everything that follows doesn't work.


Wasi: I have discovered another problem:

line 15 is O.K.

I assume p is a density, A is a cross-sectional? area and x is a length?

At any rate, assuming m,p,A are positive line 15 does not have a real solution for positive x as [sqrt(m) +sqrt(m+pAx] / [sqrt(m) - sqrt(m+pAx)] will be negative since

sqrt(m+pAx) > sqrt(m).