How about just some For loops:

```
For[y = 0, y <= 10000, y++,
For[x = 0, x <= 100, x++,
If[y^4 + 2 y^2 x^2 - 15 x == 0,
Print["My x: ", x, " my y: ", y];
];
]
]
```

or FindInstance does this but I'm not sure if we can assume the function is attempting the solutions in the order you wish.

```
FindInstance[
y^4 + 2 y^2 x^2 - 15 x == 0 && 0 <= x <= 10000 && 0 <= y <= 100, {x,
y}, Integers]
```