6.086. $\left\{\begin{array}{l}(x-y)\left(x^{2}-y^{2}\right)=45, \\ x+y=5\end{array}\right.$

Solve the system of equations:
\[
\left\{\begin{array}{l}
(x-y)\left(x^{2}-y^{2}\right)=45, \\
x+y=5
\end{array}\right.
\]
Solution.

Rewrite this system in the form

$\left\{\begin{array}{l}(x-y)(x-y)(x+y)=45, \\ x+y=5\end{array} \Leftrightarrow\left\{\begin{array}{l}(x-y)^{2}(x+y)=45, \\ x+y=5 .\end{array} \Rightarrow\right.\right.$

$\Rightarrow(x-y)^{2}=9$, from which $x-y=-3$ or $x-y=3$. We obtain a combination of two systems:

1) $\left\{\begin{array}{l}x-y=-3, \\ x+y=5,\end{array}\right.$ 2) $\left\{\begin{array}{l}x-y=3 \\ x+y=5\end{array}\right.$

Solving these systems, we find $\left\{\begin{array}{l}x_{1}=4, \\ y_{1}=1\end{array}\right.$ and $\left\{\begin{array}{l}x_{2}=1, \\ y_{2}=4 .\end{array}\right.$

Answer: $(4 ; 1)(1 ; 4)$