This paper is devoted to input-output systems modelled by differential equations. Uncertainty of the input and in the system is described by fuzzy sets. A general fuzzy framework is developed in which existence and uniqueness of a fuzzy solution is guaranteed. The approach is based on the functorial property of the extension principle. It is shown that the concept is capable of providing fuzzy solutions in place of nondifferentiable, weak solutions. The results are exemplified by means of the Burgers equation from fluid dynamics.