There's a trick, but it's easy.

$39^2=3^2\times 13^2=3^2\times (5^2+12^2)=(3\times 5)^2+(3\times 12)^2$

So, one of $x$ and $y$ is $15$ , and the other is $36$

Hence their sum is $15+36=\boxed {51}$ , a number that can be obtained by interchanging the positions of digits of one of $x$ and $y$ .

Here is the code:

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for i in range(1, 39):
    for j in range(1, 39):
        if i*i + j*j == 39*39:
            print(i, j)

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Output:
15 36
36 15

$x + y = 51$