What is the two parametric solution of the equation $\ x^2−cy−d=0 $?

Question

I'm looking for a two parametric solution of a Diophantine equation of this kind: $\ x^2−cy−d=0 $, in the set of natural numbers.

Answer 1

Are you looking for a solution of the following type?

Let $d$ be a q.r. modulo $c$. (Otherwise there is no solution, as pointed out by Mark Bennet.)

Then there is a solution $x_0^2=cy_0+d$.

Then the general solution is $$x=\pm x_0+tc,y=y_0\pm2tx_0+t^2c.$$