How many ordered pairs of positive integers $(x, y)$ satisfy the equation $\frac{x}{y} = \frac{225}{xy} + \frac{y}{x}$?

Let's solve the equation .
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Since x>0, y>0, the equation is .
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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.




All the rest combinations will provide not positive integer solutions.
(113,112), (39,36), (25,20), (17,8), (15,0)

They are somewhat similar