There are an infinite number of solutions. These solutions must obey the condition y = x^k. By applying this condition the equation x^y = y^x becomes x^x^k = x^k^x, that is x^kx = x^kx, which is true.
@AlainPaulikevitch2 ай бұрын
How do you translate what you said into an equation that gives y based on values of x? is that even possible? Using the formula from the video for x and y based on k with k > 0 and k 1 allows finding x and y while k varies. However the y = k.x form which allows this, does work but is not explained/justified while your form seems more appropriate, however it leads to a tautlogy. Deimos on android plots x^y = y^x and it seems like it should have a clean formula that i can't figure out
@Kokujou55 ай бұрын
guys... am i the only one that noticed that literally infinite solution exists? like every constellation of numbers is valid as long as y = x 2^2=2^2, 3^3=3^3, 4^4=4^4
@Tommy_0074 ай бұрын
No, you are not the only one... It is in the video!
@geraldsmith6225Ай бұрын
That's too easy
@user-lj1nd8rq9w2 ай бұрын
WTF I just saw? This one has: 1. Indefinite number of trivial solution where x=y 2. Two non-trivial integer solution x=2; y=4 and x=4; y=2 3. Indefinite number of real solution like x=3; y=2.47805268 or x=2.2; y=3.4776896 and so on. KZbin became special place where people who have no clue in subject trying to tech everyone else in this very subject. And it looks like all authors of those "mathematical" clips just learned to write and proudly show to whole world new acquired skills. Common solution. x=a^(1/(a-1)) and y=ax where a any number>0. It give us x=2, y=4 when a=2 and x=4, y=2 when a=0.5. Also there is very interesting result x=2.25, y=3.375 when a=1.5 And plethora of solutions anywhere in between.
@mohamedhamouch11223 ай бұрын
Hi; There is a typo error at 12:05. kx in the right side of the equation should be closed in parenthesis but the next is good.
@fredsmith63242 ай бұрын
you mean at 2:23? yeah, she corrected it at 3:05
@ShrikrishnaMarathe4 ай бұрын
i dont know if y=kx is a valid assumption. so it may only give partial solutions?
@sunilchaudhari9702 ай бұрын
(2,4) or (4,2) are the solutions by observation😂
@aruntiwari57145 ай бұрын
Good 😊
@abuabu24412 ай бұрын
X=y😂😂😂 X=1000 Y=1000 X^1000=y^1000 😂😂😂😂
@softlinkmangalore41332 ай бұрын
X and Y are two different variables. Two variables cannot have the same values.
@lnmukund6152Ай бұрын
When x=Ky ,thesolutions are always x=k^1/k-1, y=k^k/k-1, Mukund
@manojchaugule7945 ай бұрын
very nice🙏
@alikandis25493 ай бұрын
I have to admit it is brilliant letting k be your independent variable for one aspect, you can generate infinite solutions, also includes x=y solution. But if your aim is to assign a value for one of the variables, you cannot calculate the other. For example, at one of your infinite solutions, x=5. We know but cannot calculate y if x=5.
@vaclavkrpec28792 ай бұрын
What was the square root for, then? :-)
@arvindkulkarni65805 ай бұрын
very nice solution mam
@ericfleet96025 ай бұрын
Ummm, this is silly... any solution where x = y is going to be correct. There is no unique solution to this problem as presented.
@Tommy_0074 ай бұрын
Only children in primary school assume that an equation must have a unique solution...
@Tommy_0074 ай бұрын
The solution in the video generates infinitely many solutions.
@ericfleet96024 ай бұрын
@@Tommy_007 Cute, but stupid response. It is supposedly a Math Olympiad Question, meaning they would not give a question that is so trivial. But, only a child wouldn't realize that.
@Observer19733 ай бұрын
May you prove there are no other solutions except y=x?
@SylvainDemuyter2 ай бұрын
@@Observer1973Yes by a simple analysis of x/log(x) function.
@entertainmentzone68382 ай бұрын
X=1, Y=1 will also satisfy this inequality
@user-go5sy7mu6q3 ай бұрын
Nice
@LaxmanKumar-yi4cv5 ай бұрын
Your video is great 👍
@OneEyedJacker4 ай бұрын
I got ln(x)/sqrt(2) = ln(y)/sqrt(2) after integrating both sides, which simplifies to x=y
@vraser2 ай бұрын
Huh?? Seriously???? The answer to this is (any number, where x=y)
@stevemaskal59332 ай бұрын
no0t all negative numbers
@marklevin32363 ай бұрын
x=y. ,as long as x^x is defined.....x is either a positive number or integer negative are the solutions
@rolandkaschek97225 ай бұрын
You have assumed that x = k* y, for a constant k. That will make you miss out on many solutions! After all, as others have pointed out, (x,y)=(r,r) is a solution for each reall number r.
@Tommy_0074 ай бұрын
No. Choose k=1...
@rolandkaschek97224 ай бұрын
@@Tommy_007 thank you. Indeed I overlooked the obvious solution.
@harshnambiar3 ай бұрын
Yeah but you'd still lose all the higher power solutions like for x = ky2
@bozhidarpenkov79725 ай бұрын
How about 2^4=4^2 ???
@conigedegato26815 ай бұрын
I thought the same!!!!
@Tat523 ай бұрын
k=2, then x=2, y=kx=2*2=4
@Tat523 ай бұрын
But that equation has many solutions!
@lashaghetia87092 ай бұрын
Y=k×k^1/k-1
@valeryshebaldenkov93264 ай бұрын
И где тут russian? Решение звиздоватое... И абсолютно неверное.
@arvindkulkarni65805 ай бұрын
you are really a good teacher for math Olympiad problem
@user-wi8iq3hn3k2 ай бұрын
У нас за такое решение ставят 0 баллов из 7. Не нужно браться за задания, которые намного выше вашего арсенала.
@user-lj1nd8rq9w2 ай бұрын
Моя училка за такое решеное забила-бы меня учебником по алгебре :-)))))
@tapaskantimitra28812 ай бұрын
How you are assuming k is not equal to 1?
@sudhirsharma79662 ай бұрын
Nobody can solve 2 variables by one equation 😮
@user-lj1nd8rq9w2 ай бұрын
It is not always true. For example x+y+z=30; x^2+y^2+z^2=300 has only one solution and it is possible to show it positively. However in this particular case there are infinite number of solutions.
@uduehdjztyfjrdjciv21605 ай бұрын
x=y trivial solution and infinitely many solutions in lambert function x=y=e is both.
@WhiteGandalfs3 ай бұрын
(1:20) The square root is redundant.
@herbertklumpp29692 ай бұрын
Trivial, x=y is always a solution x,y >0
@abuabu24412 ай бұрын
0^0=0^0=>1=1
@herbertklumpp29692 ай бұрын
@@abuabu2441 only if 0^0 is well defined
@shubhrodutta2 ай бұрын
What about x=2 and Y = 4
@Tommy_0074 ай бұрын
Good video. Unfortunately, many viewers don't understand it... They don't understand that k is a parameter that generates the (infinitely many) solutions.
@Mamtamaam4 ай бұрын
Yes, you are right
@jayktomaszewski87382 ай бұрын
what about 2^4 = 4^2
@pedrofajardo81374 ай бұрын
....... Porque? Y = kx..... Nadie lo dice en el problema.
@dr.walterstadler18372 ай бұрын
Infinite number of solutions for x = y
@LaxmanKumar-yi4cv5 ай бұрын
Hello 👋
@LaxmanKumar-yi4cv5 ай бұрын
How are you ☺
@RichardSalvucci2 ай бұрын
trivial
@humelakecabin4 ай бұрын
I got X =4 and Y =2 16
@ShareTyro20102 ай бұрын
x^y= y^x , ln(x^y )=ln(y^x ), y.ln(x)=x ln(y) y/x=ln(y)/ln(x) , y/x=log_x(y) let k= y/x where k is constant obviously for k=1 it follows x=y ≠0, k =log_x(kx) k=log_x(x)-log_x(k)=1-log_x(k) k-1= log_xk =>x^(k-1)=k or x= k^(1/(k-1)), y=k^(k/(k-1)) check for k=2 x= 2^(1/(2-1))=2, y=2^(2/(2-1))=2^2=4, 2^4=4^2 or 16=16
@sanjaykamath902104 ай бұрын
0 2,4
@carmicha5 ай бұрын
But there are many other solutions, e.g., if x=y then x^y = y^x as well as 2^4=4^2 .
@arturradwanski53975 ай бұрын
The second one is actually coverd in a video, when you substitute k=2 you get x=2 & y=4
@minhhainguyen26715 ай бұрын
❤❤❤❤❤.
@davidchen2754 ай бұрын
x=y
@homeauburnRaja-hm3gq3 ай бұрын
Take log and it leads to y=x
@tomtke73514 ай бұрын
x^y = y^x clearly x, y = 1 is one solution as is x, y = 0. just by inspection in fact... any x = y solution works hmmm this author explores y = kx such that the ratio k =y/x is different than 1. x^k = xk k = x^(k-1) k^(1/(k-1))=x but y=kx so y=k[k^(1/(k-1))] so@ 7:03 in video something I don't understand happened as the author comes up with y=k^(k/(k-1)) is wrong? and I believe y=k^(2/(k-1)) is right? y log(x) = x log(y)? y/x = log(y)/log(x) or?? y÷(log(y)) = x÷(log(x)) refresher log.10(a) = b means 10^b=a
@kalyanbanerjee34902 ай бұрын
😄😄😄
@Handle-nj5fs3 ай бұрын
Time waster!!! 1:18 Couldn't you cancel the power of x from both the sides and make it x^k=kx earlier? Reached x^k=kx at 4:03. Square roots was not at all needed for that.
@davidz14363 ай бұрын
Vary complicate for a problem with a lot of solution, X = Y.
@santoshkumarpradhan52132 ай бұрын
X=Y this is the Right answer. Itina badha Chadha ke kya jarurat he
@sigillumdei8874 ай бұрын
Obviously when I have seen the thumbnail in video I realized that x=y=0. No more complications than that any number rised to power 0 is 1. This was 1 second solve then I realized there are infinite number of solutions. If this is a olimpiad question then my medium math knowledge seems advanced :))) no way can be a olimpiad one.
@ronaldnoll32473 ай бұрын
Another result is x=1 and y=1
@user-ig7nj1xb5r4 ай бұрын
i don't understand. why y=kx ( k= constant) ?
@Mamtamaam4 ай бұрын
Yes
@AlainPaulikevitch2 ай бұрын
very good question this is the most important step in the resolution and it is not even explained. it actually allows finding one family of solutions but the logic behind arbitrarily choosing it is not explained. why not try something like cos(x) = k / y (to give a silly arbitrary alternative). My point being that when solving such problems one has to justify their solution as being complete, and in this video this is not done, possibly because the trivial solution anybody can guess, however as far as i can figure there are 3 families of solutions. x = y for x and y >= 0 is a solution y = k pow(k/(k-1)) and x=k pow(1/(k-1)) for k > 0 and k not equal to 1 is another solution in the real numbers set And there is a third potential solution in the realm of complex numbers when either (or both/not sure) x or y are such that ln(x) = x or ln(y) = y (i have no idea how this is solved but googling it leads to people solving it in the complex number set) I somehow feel that the expected solution to this problem is to find an equation for the second solution which is not a parameterized equation but rather in the form y = f(x) which would be nicer, but have no reason to believe that this is possible, neither can i surmise that it isn't as not having found it does not mean it isn't there. This wouldn't be an olympiad question if the solution didn't allow for further refinements to differentiate the candidates.
@mmagic57533 ай бұрын
what if x = 1, and y = 1? r u take serious as a math tutor on youtube? dont missleading ur watcher.
@sudhirsharma79662 ай бұрын
Why not 5 not 7 hahahahahah
@mithrasrevisited48733 ай бұрын
X = 1 and Y = 1. Am I missing something?
@GeoRedtick2 ай бұрын
Valid for any x=y.
@igortchernowitzer9273 ай бұрын
2/4=4/2
@GuiAguirrezabal2 ай бұрын
Absurdo. Tanto trabalho ... Qq x=y; y sendo potencia de x ...
@xyzw24682 ай бұрын
Otra solucion es: y = x ProductLog[ - (Log[x]/x) ] )/Log[x]
@valeryshebaldenkov93264 ай бұрын
Сорри - это чисто русское решение... дебильное...
@user-zj9vt9el3w2 ай бұрын
X=4 ; Y=2
@kailasshirore222 ай бұрын
Edpat
@renebrienne18624 ай бұрын
Sorry, , but stupid ans unuseful trial of solution !!!!! Thé simple solution ( in fact an infinity)IS, of course x= y.
@rajroy443 ай бұрын
(silly) and even this silly example is wrong.
@SuperAnangs5 ай бұрын
x= 0, y = 1
@uduehdjztyfjrdjciv21605 ай бұрын
Only zero to power of zero can be one, other powers of zero are zero. One only in power of infinity can be e. Other powers of one is one.
@keescanalfp51433 ай бұрын
in your case, are we really ready to admit that 1⁰ = 0¹ so 1 = 0 . ? yeah it's almost perfect . we would not dare to write down this . but yeah you really earn your teacher's heart . good luck .