# Wolfram Language & System 11.0 (2016)|Legacy Documentation

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

# ParametricNDSolveValue

ParametricNDSolveValue[eqns,expr,{x,xmin,xmax},pars]
gives the value of expr with functions determined by a numerical solution to the ordinary differential equations eqns with the independent variable x in the range xmin to xmax with parameters pars.

ParametricNDSolveValue[eqns,expr,{x,xmin,xmax},{y,ymin,ymax}]
solves the partial differential equations eqns over a rectangular region.

ParametricNDSolveValue[eqns,expr,{x,y}Ω,pars]
solves the partial differential equations eqns over the region Ω.

ParametricNDSolveValue[eqns,expr,{t,tmin,tmax},{x,y}Ω,pars]
solves the time-dependent partial differential equations eqns over the region Ω.

## Details and OptionsDetails and Options

• ParametricNDSolveValue gives results in terms of ParametricFunction objects.
• A specification for the parameters pars of {pspec1,pspec2,} can be used to specify ranges.
• Possible forms for pspeci are:
•  p p has range Reals or Complexes Element[p,Reals] p has range Reals Element[p,Complexes] p has range Complexes Element[p,{v1,…}] p has discrete range {v1,…} {p,pmin,pmax} p has range
• Typically expr will depend on the parameters indirectly, through the solution of the differential equations, but may depend explicitly on the parameters.
• Derivatives of the resulting ParametricFunction object with respect to the parameters are computed using a combination of symbolic and numerical sensitivity methods when possible.
• ParametricNDSolveValue takes the same options and settings as NDSolve.
• NDSolve and ParametricNDSolveValue typically solve differential equations by going through several different stages, depending on the type of equations. With Method->{s1->m1,s2->m2,}, stage si is handled by method mi. The actual stages used and their order are determined by NDSolve, based on the problem to be solved.
• Possible solution stages are the same as for NDSolve, with the addition of:
•  "ParametricCaching" caching of computed solutions "ParametricSensitivity" computation of derivatives with respect to parameters

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Get a parametric function of the parameter a for the value of y:

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Evaluating with a numerical value of a gives an approximate function solution for y:

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Get the value at :

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Plot the solutions for several different values of the parameter:

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Get a function of the parameter a that gives the value of the function f at :

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This plots the value as a function of the parameter a:

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Use the function with FindRoot to find a root:

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Show the sensitivity of the solution of a differential equation to parameters:

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The sensitivity with respect to a increases with t:

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The sensitivity with respect to b does not increase with t:

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