# 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] f where 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]f partial differential equation f[u'[x],u[x],x]0 differential algebraic equation u'[x]f[x,u[x-x1]] delay differential equation integro-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 \$Assumptions assumptions on parameters DiscreteVariables {} discrete variables for hybrid equations GeneratedParameters C how to name generated parameters Method Automatic what 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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