Diophantine Equations Sx ~ 2-kxy + Y ~ 2 + Lx = 0

In this paper we investigate the solution of the Diophantine equations sx2-kxy+y2+lx=0,gcd(x,y,l)=1,(1) where k,l∈Z/{0),s∈Q/{0},we find solutions of(1)when s,k,l are in some special cases.When s=-1,(1)reduces to-x2-kxy+y2+lx=0,gcd(x,y,l)=1,(2) we showed that only when l=1,for any given k,the Diophantine equation(2) has an infinite number of positive integer solutions(x,y).For any given positive integer l>1,we prove that there are only finitely many integers k such that the Diophantine equation(2)has an infinite number of positive integer solutions (x,y).Moreover,when l=1,for some special cases sx2-kxy+y2+x=0with s,k meet some certain conditions,we give all solutions of the equations.