Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a (d)-dimensional system of linear equations or linear differential equations with complexity (\mathrm{poly}(log d)). While several of these algorithms approximate the solution to within (\epsilon) with complexity (\mathrm{poly}(log(1/\epsilon))), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity (\mathrm{poly}(log d, log(1/\epsilon))).