To be precise, I am curious about the equation
$$Ax+by=y^2$$
where all variables are positive integers and $A$ is given as some fixed constant.
I know you can do it with a Pell equation/continued fraction convergents, so to be clear, I'm wondering if there are any other ways to solve this form without brute force---I suspect there aren't, but would love to be corrected, as this is not really my area.
I've googled pretty extensively and was surprised that I couldn't find this already addressed somewhere. Thanks for any help!