Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

DSolve

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

DSolve[eqn,u,{x,xmin,xmax}]
solves a differential equation for x between and .

DSolve[{eqn1,eqn2,},{u1,u2,},]
solves a list of differential equations.

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

DSolve[eqn,u,{x1,x2,}Ω]
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]f partial differential equation f[u'[x],u[x],x]0 differential 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 \$Assumptions assumptions on parameters DiscreteVariables {} discrete variables for hybrid equations GeneratedParameters C how to name generated parameters Method Automatic what 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][].  »

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

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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Substitute the solution into an expression:

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