We consider perturbations of the problem (*) \(-x''+bx=\lambda ax,x(0)- x(1)=0, x'(0)-x'(1)=0\) both by changes of the boundary conditions and by addition of nonlinear terms. We assume that at \(\lambda =\lambda_ 0\) there are two linearly independent solutions of (*) and that a is bounded away from zero. When only the boundary conditions are perturbed either the Hill's discriminant or the method of Lyapunov-Schmidt reduces the problem to \(0=\det((\lambda -\lambda_ 0)A-\epsilon H)\) plus higher order terms, where A and H are real constant 2\(\times 2\) matrices. We analyze the existence of curves (\(\lambda\) (\(\epsilon)\),\(\epsilon)\) of eigenvalues for this problem of linear perturbation and give examples. The method of Lyapunov-Schmidt is used to analyse the nonlinear problem. In a sequel to this paper we will analyze examples of the bifurcation problem where the perturbations preserve all, some, or none of the symmetry of the linear unperturbed problem (*). When all of the symmetry is broken we can use generic bifurcation techniques, and when some or all of the symmetry is preserved we can use the symmetry of the bifurcation equations.