Is there an analytic solution for equation $4x^2y(x^2y-x+1)=15x^2+2x-1$?
The integer solutions can be found by reasoning; clearly we mus t have:
$15x^2+2x-1=4 k$
$\Delta'=1+15(1+4k)=16+60 k= t^2$
$(t-4)(t+4)= 3 k \times 20$
$t+4=20$ ⇒ $t=16$
⇒ $3k=16-4=12$ ⇒ $k=4$
⇒ $x= 1$
Putting in equation we get $y=2$
Can these type of equations have analytic solutions, especially when we want all solutions in $\mathbb {R}$?
$$4x^2y(x^2y-x+1)-15x^2-2x+1=4x^4y^2-4x^3y+4x^2y-(5x-1)(3x+1)=$$ $$4x^4y^2-2x^2y(5x-1-(3x+1))-(5x-1)(3x+1)=(2x^2y-5x+1)(2x^2y+3x+1).$$ Can you end it now?