\[x^{2}(x - y) \geq y^{2}(x - y);\ \ \ \ \ \ \]
\[x \geq 0;\ \ y \geq 0\]
\[(x - y)\left( x^{2} - y^{2} \right) \geq 0\]
\[(x - y)(x - y)(x + y) \geq 0\]
\[(x - y)^{2}(x + y) \geq 0\]
\[(x - y)^{2} > 0\ \ всегда;\ \ \]
\[x + y \geq 0 - из\ условия.\]