We study nonnegative integer solutions of the Diophantine equation Ax + By=z2, where A ≡ 3 (mod 20) with the condition that (A + 1) is a perfect square and B ≡ 5 (mod 20) are positive integers and prove that the equation has a unique solution (x, y, z)=(Formula Presented). Then we present a full study of positive integer solutions (p, x, y, z) of the equation px + py=2z2, where p=2 and p ≡ 3, 5 (mod 8) are prime numbers. Finally, we present a detailed analysis of positive integer solutions (p, x, y, z) of px − py=2z2, when p ≥ 2 is a prime. At the end, we provide few tables consisting of computational evidences corroborating our results.