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Documentation / Mathematica / Built-in Functions / Algebraic Computation / Calculus /

DSolve

FilledSmallSquare DSolve[eqn, y, x] solves a differential equation for the function y, with independent variable x.

FilledSmallSquare DSolve[, , ... , , , ... , x] solves a list of differential equations.

FilledSmallSquare DSolve[eqn, y, , , ... ] solves a partial differential equation.

FilledSmallSquare DSolve[eqn, y[x], x] gives solutions for y[x] rather than for the function y itself.

FilledSmallSquare Example: DSolve[y'[x] == 2 a x, y[x], x] LongRightArrow.

FilledSmallSquare Differential equations must be stated in terms of derivatives such as y'[x], obtained with D, not total derivatives obtained with Dt.

FilledSmallSquare The list of equations given to DSolve can include algebraic ones that do not involve derivatives.

FilledSmallSquare DSolve generates constants of integration indexed by successive integers. The option GeneratedParameters specifies the function to apply to each index. The default is GeneratedParameters->C, which yields constants of integration C[1], C[2], ... .

FilledSmallSquare GeneratedParameters->(Module[{C}, C]&) guarantees that the constants of integration are unique, even across different invocations of DSolve.

FilledSmallSquare For partial differential equations, DSolve generates arbitrary functions C[n][... ].

FilledSmallSquare Boundary conditions can be specified by giving equations such as y'[0] == b.

FilledSmallSquare Solutions given by DSolve sometimes include integrals that cannot be carried out explicitly by Integrate. Dummy variables with local names are used in such integrals.

FilledSmallSquare DSolve sometimes gives implicit solutions in terms of Solve.

FilledSmallSquare DSolve can solve linear ordinary differential equations of any order with constant coefficients. It can solve also many linear equations up to second order with non-constant coefficients.

FilledSmallSquare DSolve includes general procedures that handle a large fraction of the nonlinear ordinary differential equations whose solutions are given in standard reference books such as Kamke.

FilledSmallSquare DSolve can find general solutions for linear and weakly nonlinear partial differential equations. Truly nonlinear partial differential equations usually admit no general solutions.

FilledSmallSquare DSolve can handle not only pure differential equations but also differential-algebraic equations.

FilledSmallSquare See Section 1.5.9 and Section 3.5.10.

FilledSmallSquare Implementation Notes: see Section A.9.5.

FilledSmallSquare See also: NDSolve, Solve, RSolve.

FilledSmallSquare Related packages: Calculus`VariationalMethods`, Calculus`VectorAnalysis`.

FilledSmallSquare New in Version 2; modified in 5.0.

Further Examples