Let's solve the equation [tex] \frac{x}{y} = \frac{225}{xy} + \frac{y}{x} [/tex].
[tex] \frac{x}{y} = \frac{225+y^2}{xy} [/tex].
Since x>0, y>0, the equation is [tex]x= \frac{225+y^2}{x} \Rightarrow x^2=225+y^2\Rightarrow x^2-y^2=225[/tex].
[tex](x-y)(x+y)=3\cdot3\cdot5\cdot5[/tex].
The divisors of 225 are: 1, 3, 5, 9, 15, 25, 45, 75, 225. Notice that x-y<x+y, that's why all positive integer solutions of the equation are solutions of the next systems.
[tex] \left \{ {{x-y=1} \atop {x+y=225}} \right. \rightarrow \left \{ {{x=113} \atop {y=112}} \right. [/tex]
[tex] \left \{ {{x-y=3} \atop {x+y=75}} \right. \rightarrow \left \{ {{x=39} \atop {y=36}} \right. [/tex]
[tex] \left \{ {{x-y=5} \atop {x+y=45}} \right. \rightarrow \left \{ {{x=25} \atop {y=20}} \right. [/tex]
[tex] \left \{ {{x-y=9} \atop {x+y=25}} \right. \rightarrow \left \{ {{x=17} \atop {y=8}} \right. [/tex]
All the rest combinations will provide not positive integer solutions.
Answer: (113,112), (39,36), (25,20), (17,8), (15,0)
