Existence of solutions to $a^2x^2 + bx = y^2$

Question

Is there a way to determine if there are or aren't any integer positive solutions $(x,y)$ to the equation $a^2x^2 + bx = y^2$ depending on the values of $a$ and $b$?

I tried to deal with it using Pell equations but I just couldn't work it out. Any help is appreciated.

Answer 1

Above equation shown below has solution:

$a^2x^2 + bx = y^2$

$x=3(k-3)^2$

$y=3(k-3)(3k-4)$

$a=2$

$b=15(k^2-4)$

For $k=5$ we get:

$(x,y,a,b)= (12,66,2,315)$

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Answer 2

"OP" has follow up question for equation shown below:

$a^2x^2+bx=y^2+w$

For w=2^6 & w=2^12 there are solutions shown below:

$x=k^2-10k+29$

$y=5k^2-33k+20$

$a=2$

$b=25(k^2-4)$

$w=(2k^2-8)^2$

For k=6 we get w= 2^12 and :

$(x,y,a,b)= (5,2,2,800)$

For k=0 we get w =2^6 and,

$(x,y,a,b)= (29,20,2,-100)$