It is clear that $y$ is an odd number. Hence $y^2 \equiv 1 \pmod{8}$ and $y^2 \equiv 1 \pmod{4}$. Rewriting the given equation as $$2x^2 = 5y^2 + 7$$

If $x$ is an even number, then $2x^2 \equiv 0 \pmod{8}$. But $5y^2 + 7 \equiv 5+7 \equiv 4 \pmod{8}$. It is a contradiction.

If $x$ is an odd number, then $2x^2 \equiv 2 \pmod{4}$. But $5y^2 + 7 \equiv 0 \pmod{4}$. It is a contradiction too.