DSolveValue

DSolveValue[eqn,expr,x]

gives the value of expr determined by a symbolic solution to the ordinary differential equation eqn with independent variable x.

DSolveValue[eqn,expr,{x,xmin,xmax}]

uses a symbolic solution for x between xmin and xmax.

DSolveValue[{eqn1,eqn2,},expr,]

uses a solution for the partial differential equation eqn.

DSolveValue[eqn,expr,{x1,x2,}]

uses a symbolic solution for a list of differential equations.

DSolveValue[eqn,expr,{x1,x2,}Ω]

uses a solution of the partial differential equation eqn over the region Ω.

Details and Options

  • DSolveValue can solve ordinary differential equations (ODEs), partial differential equations (PDEs), differential algebraic equations (DAEs), delay differential equations (DDEs), integral equations, integro-differential equations and hybrid differential equations.
  • The output from DSolveValue is controlled by the form of the dependent function, u or u[x]:
  • DSolveValue[eqn,u,x]fwhere f is a pure function
    DSolveValue[eqn,u[x],x]f[x]where f[x] is an expression in x
  • DSolveValue will return a single solution, whereas DSolve can return multiple solutions.
  • DSolveValue can give solutions that include sums and integrals that cannot be carried out explicitly. Variables K[1], K[2], are used in such cases.
  • DSolveValue[eqn,y[Infinity],x] gives the limiting value of the solution y at Infinity.
  • Different classes of equations solvable by DSolveValue 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
    u'[x]+int_(a)^(b)k[x,t]u[t]dt⩵fintegro-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 u[x1]a, u'[x2]b, etc.
  • Boundary conditions for PDEs can be given as equations u[x,y1]a, 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 g[x] in the form u[x/;x<x0]g[x].
  • 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[DSolveValue[...]] calls NDSolveValue or ParametricNDSolveValue 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 control 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][].  »

Examples

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Basic Examples  (3)

Solve a differential equation:

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Include a boundary condition:

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Get a "pure function" solution for :

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Plot the solution:

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Obtain the value of the solution at a point:

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Derivative of the solution:

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Scope  (99)

Generalizations & Extensions  (2)

Options  (5)

Applications  (31)

Properties & Relations  (9)

Possible Issues  (2)

Neat Examples  (2)

See Also

DSolve  NDSolveValue  AsymptoticDSolveValue  WhenEvent  DEigensystem  DEigenvalues  NDEigensystem  NDEigenvalues  GreenFunction  Solve  RSolve  Integrate  DifferentialRoot  StreamPlot  ItoProcess  SystemModelSimulate

Tutorials

Introduced in 2014
(10.0)
| Updated in 2017
(11.2)