Solving x^y=y^x

10242018, 06:06 PM
(This post was last modified: 10242018 06:07 PM by sasa.)
Post: #1




Solving x^y=y^x
This guy have quite interesting samples to play with...
Is it possible to solve this one on Prime? 

10242018, 07:21 PM
Post: #2




RE: Solving x^y=y^x
solve(subst(x^y=y^x, y=t*x), x) ==> [ t^(1/(t1)) ]
So, [x, y] = [ t^(1/(t1)), t^(t/(t1)) ] But, I may have discovered a XCas bug ... solve([x^y=y^x, y=t*x], x) ==> *crash* 

10242018, 07:37 PM
(This post was last modified: 10242018 07:38 PM by Dieter.)
Post: #3




RE: Solving x^y=y^x
(10242018 06:06 PM)sasa Wrote: This guy have quite interesting samples to play with... I didn't watch the video, but this equation can be directly solved by means of the Lambert W function. The WP34s, for example, handles this easily: LN RCL/ L CHS Wp RCL/ L Enter x, get y and vice versa. Replace Wp with Wm (the other branch of Lambert's W function) and get the second solution. For x=e both solutions are equal (x=y=e) X=2 => Y=2 X=3 => Y=2,47805268... X=4 => Y=2 So, does the Prime feature the W function? Dieter 

10242018, 07:45 PM
(This post was last modified: 10242018 07:46 PM by sasa.)
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RE: Solving x^y=y^x  
10242018, 08:23 PM
(This post was last modified: 10242018 08:24 PM by John Keith.)
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RE: Solving x^y=y^x  
10242018, 08:45 PM
(This post was last modified: 10242018 08:46 PM by sasa.)
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RE: Solving x^y=y^x  
10242018, 09:58 PM
Post: #7




RE: Solving x^y=y^x
(10242018 07:37 PM)Dieter Wrote: The WP34s, for example, handles this easily: Thanks, Dieter. I was curious how above get derived ... Found a very close solution for infinite tetration c = x^c, or ln(c)/c = ln(x). From the video (~ 7:00), using basic algebra, he got c = W(ln(x)) / (ln(x)) For x^y = y^x, ln(y)/y = ln(x)/x, so just substitute above: c=y, ln(x)=ln(x)/x > y = W(ln(x)/x) / (ln(x)/x) 

10242018, 10:35 PM
(This post was last modified: 10242018 10:37 PM by ijabbott.)
Post: #8




RE: Solving x^y=y^x
(10242018 09:58 PM)Albert Chan Wrote: Found a very close solution for infinite tetration c = x^c, or ln(c)/c = ln(x). I like his series of videos on geometric algebra  something that isn't currently handled very well by our favourite, handheld, symbolic calculators. — Ian Abbott 

10262018, 04:54 PM
Post: #9




RE: Solving x^y=y^x
(10242018 10:35 PM)ijabbott Wrote:(10242018 09:58 PM)Albert Chan Wrote: Found a very close solution for infinite tetration c = x^c, or ln(c)/c = ln(x). Hallo, The question in the video for Infinite Tetration was the point for me to put out my old calculator. So I can only show a solution for Infinite Tetration on the HP71b to see if the number converges. Define the WFunktion y=x*exp(x) and put the formula on minute 12:10 in the keyboard. Looks like like this: Definiton of the Wfunction in BASIC 10 DEF FNW(X) = FNROOT(0,10,FVAR*EXP(FVAR)X) and then his example with 1.4 on the keyboard calling there defined Wfunction: FNW((LN(1.4)))/(LN(1.4)) Result: 1.88666330624 and then his example with SQR(2) on the keyboard calling there defined Wfunction: FNW((LN(SQR(2)))/(LN(SQR(2))) Result: 1.99999999998 not exactly 2 but this seems is a limitation on the more then 30 year old calculator. The possibility for defining a function and using it straight from the keyboard was a revolution in this times. best regards Erwin 

10282018, 08:21 PM
(This post was last modified: 10282018 08:25 PM by Albert Chan.)
Post: #10




RE: Solving x^y=y^x
(10262018 04:54 PM)Erwin Wrote: Definiton of the Wfunction in BASIC Unless W function is builtin, it might be easier solve directly for infinite tetration. So, for any x, solve for y, such that x^y = y Below solved with my Casio FX115MS solver, using x as initial guess for y: For x=1.4, y = 1.886663306 For x= √2, y = 2 For x=1.5, y does not converge Last x > e^(1/e), thus diverged. see Exponential Reiterated 

10282018, 08:22 PM
Post: #11




RE: Solving x^y=y^x
(10242018 08:23 PM)John Keith Wrote:(10242018 07:37 PM)Dieter Wrote: So, does the Prime feature the W function? If it's not in KhiCAS then it's not coming. Completeness compared to other CAS demands a full implementation of Lambert W including (i)laplace, solve, int, diff eq's, etc. I hope there are no GPL problems... 

10282018, 08:51 PM
Post: #12




RE: Solving x^y=y^x
(10282018 08:21 PM)Albert Chan Wrote: Unless W function is builtin, it might be easier solve directly for infinite tetration.[...]Of course you are right  its much easier in your case. The HP71b has the possibility to define systemwide new functions and use it in Calculator and BASIC mode (of course FORTH too)  that was the reason why I grab my calculator to try it :) 

10292018, 03:14 AM
Post: #13




RE: Solving x^y=y^x
(10282018 08:51 PM)Erwin Wrote: possibility to define systemwide new functions and use it in Calculator and BASIC mode (of course FORTH too)  that was the reason why I grab my calculator to try it :) Prime does the same thing  either through a program or a simple define a function interface. The issue purely lies in tying that into the inner workings of the CAS to be able to recognize when it should use that function in simplifications, integrals, etc. They really are two separate things. TW Although I work for HP, the views and opinions I post here are my own. 

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