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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line to any desired level of accuracy. The solution of a DAE consists of two parts, first the search for consistent initial values and second the computation of a trajectory. To find consistent initial values it is often necessary to consider the derivatives of some of the component functions of the DAE. The highest order of a derivative that is necessary in this process is called the differentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory.