This paper is concerned with the numerical solution of singularly- perturbed boundary value problems of type: \[ (1)\quad (\epsilon +px)^ qu''+a(x)u'-f(x,u)=0,\quad 0<x<1, \] u(0)\(=u_ 0\), \(u(1)=u_ 1\) where \(\epsilon\) is a small parameter, \(p=0,1\), \(a(x)\in C^ 2[0,1]\), \(f(x,u)\in C^ 2([0,1]\times {\mathbb{R}})\), with \(f_ u>0\) and q is a positive integer. First, using some techniques developed by the first author and N. N. Yanenko [Proc. Third Int. Conf. Boundary and Interior Layers - Computational and Asymptotic Methods, Dublin 1984, 68- 80 (1984)], estimates of the derivatives of u(x) for different values of p and q are stated. In particular a function \(\phi\) (x), depending on the coefficients of (1), is defined so that \(| u'(x)| \leq \phi (x)\). Then, with such an estimate a nonuniform mesh \(x_ i=x(ih),\) \(i=0,...,N\) is constructed by means of a mapping x(q) defined by \(q=M^{- 1}\int^{x}_{0}\phi (x)dx\) with \(M=\int^{1}_{0}\phi (x)dx.\) Finally, a standard finite difference scheme (with order \(\geq 1)\) is proposed to solve (1) on this nonuniform mesh and some numerical examples are presented.