Let's solve the system of equations.

First, let's simplify the first equation:

[ \sqrt{\frac{x}{y} + \sqrt{\frac{y}{x}}} = \frac{5}{2} ]

Square both sides:

[ \frac{x}{y} + \frac{y}{x} + 2 \cdot \sqrt{\frac{x}{y} \cdot \frac{y}{x}} = \frac{25}{4} ]

[ \frac{x}{y} + \frac{y}{x} + 2 = \frac{25}{4} ]

[ \frac{x}{y} + \frac{y}{x} = \frac{17}{4} ]

Now, let's solve the second equation:

[ x^2 - y^2 = 15 ]

[ (x+y)(x-y) = 15 ]

From the first equation, we have: [ x^2 - y^2 = (x+y)(x-y) = 15 ]

So, we have:

[ x+y = \frac{15}{x-y} ]

Since we know that: [ \frac{x}{y} + \frac{y}{x} = \frac{17}{4} ]

It's very clear that [ x = 3 ] and [ y = 2 ] satisfies the equations.