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# Differential-Algebraic Equation

A differential-algebraic equation (DAE) is a type of differential equation in which the derivatives are not (in general) expressed explicitly, and typically derivatives of some of the dependent variables may not appear in the equations at all. The general form of a system of DAEs is given by

where . Differential-algebraic equations can be solved numerically in the Wolfram Language using the command NDSolve, and some can be solved exactly with DSolve.

A system of DAEs can be converted to a system of ordinary differential equations by differentiating it with respect to the independent variable .

Delay Differential Equation, Differential Algebra, Ordinary Differential Equation, Universal Differential Equation

Portions of this entry contributed by Rob Knapp

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## References

Ascher, U. and Petzold, L. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia, PA: SIAM Press, 1998.Brenan, K.; Campbell, S.; and Petzold, L. Numerical Solutions of Initial-Value Problems in Differential-Algebraic Equations. New York: Elsevier, 1989. Brown, P. N.; Hindmarsh, A. C.; and Petzold, L. R. "Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems." SIAM J. Sci. Comput. 15, 1467-1488, 1994.Brown, P. N.; Hindmarsh, A. C.; and Petzold, L. R. "Consistent Initial Condition Calculation for Differential-Algebraic Systems." SIAM J. Sci. Comput. 19, 1495-1512, 1998.Deuflhard, P.; Hairer, E. and Zugck, J. "One-Step and Extrapolation Methods for Differential-Algebraic Systems." Numer. Math. 51, 501-516, 1987.Hairer, E. and Lubich, C. On Extrapolation Methods for Stiff and Differential-Algebraic Equations. Leipzig, Germany: Teubner, 1988.Hairer, E. and Wanner, G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd ed. Berlin: Springer-Verlag, 1996.Hindmarsh, A. and Taylor, A. "User Documentation for IDA: A Differential-Algebraic Equation Solver for Sequential and Parallel Computers." Lawrence Livermore National Laboratory report, UCRL-MA-136910, December 1999.

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Differential-Algebraic Equation

## Cite this as:

Knapp, Rob and Weisstein, Eric W. "Differential-Algebraic Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Differential-AlgebraicEquation.html