The positive integer solutions \((x, y, z)\) of some special forms of simultaneous Diophantine equations \(ax^2 - bz^2 = \delta_1, cy^2 - dz^2 = \delta_2\) are investigated. Here \(a, b, c\) and \(d\) are positive integers, \(\delta_1\) and \(\delta_2\) are integers such that \(\gcd(ab, \delta_1) = \gcd(cd, \delta_2) = 1\). Also discussed are the special Diophantine equations \(x^2 - az^2 = y^2 - bz^2 = 1\), where \(a\) and \(b\) are distinct positive integers. Elliptic curves and the theory of linear forms in logarithms to effectively bound all solutions \((x, y, z)\) are applied. A result of \textit{A. Baker} and \textit{G. Wüstholz} [J. Reine Angew. Math. 442, 19--62 (1993; Zbl 0788.11026)] is frequently used.