to (abstract) algebra.] 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 for 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. A semi-explicit DAE system can be converted to an implicit one by decreasing the differentiation index by one, and vice versa. Other forms of DAEs The distinction of DAEs to ODEs becomes apparent if some of the dependent variables occur without their derivatives. The vector of dependent variables may then be written as pair and the system of differential equations of the DAE appears in the form where , a vector in , are dependent variables for which derivatives are present (differential variables), , a vector in , are dependent variables for which no derivatives are present (algebraic variables), , a scalar (usually time) is an independent variable. is a vector of functions that involve subsets of these variables and derivatives. As a whole, the set of DAEs is a function Initial conditions must be a solution of the system of equations of the form Examples The behaviour of a pendulum of length L with center in (0,0) in Cartesian coordinates (x,y) is described by the Euler–Lagrange equations where is a Lagrange multiplier. The momentum variables u and v should be constrained by the law of conservation of energy and their direction should point along the circle. Neither condition is explicit in those equations. Differentiation of the last equation leads to restricting the direction of motion to the tangent of the circle. The next derivative of this equation implies and the derivative of that last identity simplifies to which implicitly implies the conservation of energy since after integration the constant is the sum of kinetic and potential energy. To obtain unique derivative values for all dependent variables the last equation was three times differentiated. This gives a differentiation index of 3, which is typical for constrained mechanical systems. If initial values and a sign for y are given, the other variables are determined via , and if then and . To proceed to