Find integers $x$ and $y$ such that $$x^3+x-y^2=1.$$
$$x^3+x-y^2=1 \implies x^3+x-1=y^2.$$
Now, when $x^3+x-1$ is a perfect square?
The elliptic curve $y^2=x^3+x-1$ has only finitely many integral points, according to the magma online calculator - see here at MO, namely the points $$ (x,y)=(1,\pm 1),(2,\pm3),(13,\pm 47). $$