The Diophantine equation $x^4+1=Dy^2$

An effective method is derived for solving the equation of the title in positive integers $x$ and $y$ for given $D$ completely, and is carried out for all $D<100000$. If $D$ is of the form $m^4+1$, then there is the solution $x=m$, $y=1$; in the above range, except for $D=70258$ with solution $x=261$, $y=257$, there are no other solutions.