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Integrate (modified)Sum (modified)

DSolve

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

FilledSmallSquareDSolve[, , ... , , , ... , x] solves a list of differential equations.

FilledSmallSquareDSolve[eqn, y, , , ... ] solves a partial differential equation.

FilledSmallSquareDSolve[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.

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

FilledSmallSquareDSolveConstants->(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.

FilledSmallSquareDSolve sometimes gives implicit solutions in terms of Solve.

FilledSmallSquareDSolve 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.

FilledSmallSquareDSolve 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.

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

FilledSmallSquare See The Mathematica Book: Section 1.5.8 and Section 3.5.10.

FilledSmallSquare Implementation Notes: see section A.9.5.

FilledSmallSquare See also: NDSolve, Solve.

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

Further Examples

Integrate (modified)Sum (modified)