10.1. Given the equation $x^{3}+5 y=y^{3}+5 x$. Do there exist solutions to this equation for a) natural numbers $x \neq y$? ; b) positive real numbers $x \neq y$?
Answer: a) do not exist; b) exist. Solution. a) See item a) of problem 9.1. b) Let's take, for example, $y=2 x$, and substitute it into the last equation: $x^{2}+2 x^{2}+4 x^{2}=5$. Then $x=\sqrt{\frac{5}{7}}$ and $y=2 \sqrt{\frac{5}{7}}$ satisfy the equation.