$2^x+3^y=5^{xy}$

Clearly $x=y=1$ satisfy the relation

Take $x>y$

$3^x>3^y$

$2^x+3^x>2^x+3^y$

$2^x+3^x>5^{xy}$

this is false

$5^{xy}>5^x={\left(2+3\right)}^x>2^x+3^x$

Similarly there is no solution for $x<y$

Hence $x=y$ Hence only $x=y=1$ satisfy the given equation ($2^x+3^x=5^{xy}$ is not true for $x\in N-\left\{1\right\}$)