In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations \t\t\ta0(t)Dβ2y(t)+a1(t)Dβy(t)+a2(t)y(t)=b(t),t∈I,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$a_{0}(t)D_{\\beta}^{2}y(t)+a_{1}(t)D_{\\beta}y(t)+a_{2}(t)y(t)=b(t),\\quad t \\in I, $$\\end{document}a_{0}(t)neq0, in a neighborhood of the unique fixed point s_{0} of the strictly increasing continuous function β, defined on an interval Isubseteq{mathbb{R}}. These equations are based on the general quantum difference operator D_{beta}, which is defined by D_{beta}{f(t)}= (f(beta(t))-f(t) )/ (beta(t)-t ), beta(t)neq t. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.