Question 157633
You can iterate to find solutions.
Start with x=0.
{{{x^y+y^x=1061}}}
{{{0+y^0=1061}}}
{{{1=1061}}}
No solution at x=0.
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Next x=1/2.
{{{x^y+y^x=1061}}}
{{{(1/2)^y+sqrt(y)=1061}}}
Actually, y becomes large and the {{{(1/2)^y}}} term goes to zero, so the equation looks like,
{{{sqrt(y)=1061}}} and
y=1061*1061=1125721
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Next x=1. That's an easy one.
{{{x^y+y^x=1061}}}
{{{1^y+y^1=1061}}}
{{{1+y=1061}}}
{{{y=1060}}}
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Next x=2,
{{{x^y+y^x=1061}}}
{{{2^y+y^2=1061}}}
Here's where the iteration takes place.
I set up an EXCEL spreadsheet to home in on a solution.
{{{y=-32.572}}} and
{{{y=9.911}}}
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Next x=3,
{{{x^y+y^x=1061}}}
{{{3^y+y^3=1061}}}
Here's where the iteration takes place.
I set up an EXCEL spreadsheet to home in on a solution.
{{{y=6.12}}}
Looks like there's only one solution.
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(1/2,1125721)
(1,1060)
(2,-32.572)
(2,9.911)
(3,6.12)
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Hope it helps. 
Good luck.