Ta có:
$P=\frac{x^2}{y^2}+\frac{y^2}{x^2}-3(\frac{x}{y}+\frac{y}{x})+5$
$=(\frac{x}{y}+\frac{y}{x})^2-3(\frac{x}{y}+\frac{y}{x})+3$
$=((\frac{x}{y}+\frac{y}{x})^2-2(\frac{x}{y}+\frac{y}{x}).\frac{3}{2}+\frac{9}{4}+\frac{3}{4}$
$=(\frac{x}{y}+\frac{y}{x}-\frac{3}{2})^2+\frac{3}{4}≥\frac{3}{4}$
Vậy $GTNN$ của $P$ bằng $\frac{3}{4}$.