DSolve

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

DSolve[{eqn1, eqn2, ...}, {y1, y2, ...}, x]
solves a list of differential equations.

DSolve[eqn, y, {x1, x2, ...}]
solves a partial differential equation.

Details and OptionsDetails and Options

  • DSolve[eqn, y[x], x] gives solutions for rather than for the function y itself.
  • Differential equations must be stated in terms of derivatives such as , 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 typically generates arbitrary functions C[n][...]. »
  • Boundary conditions can be specified by giving equations such as .
  • 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 also solve many linear equations up to second order with nonconstant coefficients.
  • DSolve includes general procedures that handle almost all 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. »
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