Expert: Scott A Wilson - 7/22/2008

Question
QUESTION: Solve the following equations for x and y:

x + y = 5

x^y + y^x = 17

ANSWER: Try 0 and 5.  Try 1 and 4.  Try 2 and 3.  Those are the only solutions that could work since x^y + y^x is an integer.

By the way, 0^5 is still 0 and 5^0 is 1.  0+1 != 17, so that choice is no good.

---------- FOLLOW-UP ----------

QUESTION: Dear Scott,

I was looking for a mathematical solution, not a solution by trial and error. If you can help.

Shafiq

Answer
One of those was the exact solution - there's only two number to try.
1 and4 or 2 and 3.

1+4=5, 1^4 + 4^1 = 1+4 = 5, so that one doesn't work.

The other one is 2+3=5, 2^3 + 3^2 = 8 + 9 = 17 - that's it!

That is the answer
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*** x=2, y=3 ***
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In case you're curious:

Trying to find the solution exactly is almost impossible when one of the variables is in the exponet.  If the integers didn't work, I'm not sure what could be done to find the solution except by using some form of the bisection method, the secant method, or Newton's method.

For example, if that 17 was changed to an 18, the question would really be a stinker.

To solve it, I used the binomial method and Excel.
Here are the results
2.1          2.9         17.95359028
2.2          2.8        18.72728987
2.15        2.85       18.36454346
2.125       2.875      18.16490437
2.1125      2.8875     18.06068191
2.10625     2.89375    18.00749156
2.103125   2.896875   17.98062939
2.1046875   2.8953125   17.99408264
2.10546875   2.89453125   18.00079265
2.105078125   2.894921875   17.99743903
2.105273438   2.894726563   17.99911619
2.105371094   2.894628906   17.99995451
2.105419922   2.894580078   18.0003736
2.105395508   2.894604492   18.00016406
2.105383301   2.894616699   18.00005928
2.105377197   2.894622803   18.0000069
2.105374146   2.894625854   17.9999807
2.105375671   2.894624329   17.9999938
2.105376434   2.894623566   18.00000035
2.105376053   2.894623947   17.99999707
2.105376244   2.894623756   17.99999871
2.105376339   2.894623661   17.99999953
2.105376387   2.894623613   17.99999994
2.10537641   2.89462359   18.00000014
2.105376399   2.894623601   18.00000004
2.105376393   2.894623607   17.99999999
2.105376396   2.894623604   18.00000001
2.105376394   2.894623606   18

That still may not be exact, but close enough for spreadsheet purposes.  The results were 2.105376394   and 2.894623606, whose sum is 5 and whose powers satisfied a^b + b^a = 18.

That's probably the shortest method, but that's the easiest.

The secant method, which is the second hardest, is much faster.  It goes like this:

2.1   2.9   17.95359028
2.2   2.8   18.72728987
2.105998417   2.894001583   18.00533555
2.105303705   2.894696295   17.99937604
2.105376442   2.894623558   18.00000041
2.105376394   2.894623606   18

I won't even try Newton's method, but it x(n+1)=x(n)-f(x(n))/f'(x(n))
Note that equation would be f(x)=x^(5-x)+(5-x)^x.  Now that's got a complicated derivative with two product rules and two exponentials.