solves a differential equation for the function u, with independent variable x.

solves a differential equation for x between and .

solves a list of differential equations.

solves a partial differential equation.

solves the partial differential equation eqn over the region Ω.

Details and OptionsDetails and Options

  • DSolve can solve ordinary differential equations (ODEs), partial differential equations (PDEs), differential algebraic equations (DAEs), delay differential equations (DDEs), and hybrid differential equations.
  • The output from DSolve is controlled by the form of the dependent function u or :
  • DSolve[eqn,u,x]where f is a pure function
    DSolve[eqn,u[x],x]where is an expression in x
  • With a pure function output, can be used to verify the solution.  »
  • DSolve can give implicit solutions in terms of Solve.  »
  • DSolve can give solutions that include sums and integrals that cannot be carried out explicitly. Variables , , are used in such cases.
  • Different classes of equations solvable by DSolve include:
  • u'[x]f[x,u[x]]ordinary differential equation
    a xu[x,y]+b yu[x,y]fpartial differential equation
    f[u'[x],u[x],x]0differential algebraic equation
    u'[x]f[x,u[x-x1]]delay differential equation
    {,WhenEvent[cond,u[x]g]}hybrid differential equation
  • Boundary conditions for ODEs and DAEs can be specified by giving equations at specific points such as , , etc.
  • Boundary conditions for PDEs can be given as equations , Derivative[1,0][u][x,y1]b, etc. or as DirichletCondition[u[x,y]g[x,y],cond].
  • Initial conditions for DDEs can be given as a history function in the form .
  • WhenEvent[event,action] may be included in the equations eqn to specify an action that occurs when event becomes True.
  • The region Ω can be anything for which RegionQ[Ω] is True.
  • N[DSolve[]] calls NDSolve or ParametricNDSolve for differential equations that cannot be solved symbolically.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DiscreteVariables{}discrete variables for hybrid equations
    GeneratedParametersChow to name generated parameters
    MethodAutomaticwhat method to use
  • GeneratedParameters controls the form of generated parameters; for ODEs and DAEs these are by default constants C[n] and for PDEs they are arbitrary functions C[n][].  »
Introduced in 1991
| Updated in 2015
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