DSolve
 Usage
• DSolve[eqn, y, x] solves a differential equation for the function y, with independent variable x.
• DSolve[{ ,  , ... }, { ,  , ... }, x] solves a list of differential equations.
• DSolve[eqn, y, { ,  , ... }] solves a partial differential equation.
Notes
• DSolve[eqn, y[x], x] gives solutions for y[x] rather than for the function y itself. • Example: DSolve[y'[x]  2 a x, y[x], x]  . • Differential equations must be stated in terms of derivatives such as y'[x], obtained with D, not total derivatives obtained with Dt. • The list of equations given to DSolve can include algebraic ones that do not involve derivatives. • 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], ... . • GeneratedParameters->(Module[{C}, C]&) guarantees that the constants of integration are unique, even across different invocations of DSolve. • For partial differential equations, DSolve generates arbitrary functions C[n][ ... ]. • Boundary conditions can be specified by giving equations such as y'[0] b. • 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. • DSolve sometimes gives implicit solutions in terms of Solve. • 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. • 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. • DSolve can find general solutions for linear and weakly nonlinear partial differential equations. Truly nonlinear partial differential equations usually admit no general solutions. • DSolve can handle not only pure differential equations but also differential-algebraic equations. • New in Version 2; modified in 5.
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